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Accuracy of Approximations for Autoregressive Processes
Quality of Approximations
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
• Approximations may not be good when integrating in high-dimensional space (large
n)
• Relative error is more informative than absolute error for comparing tails
• Plots use logarithmic scale for both axes to compare 𝑃𝛾𝑛(𝑘)> 𝑡 with theoretical
approximation
Simulation Parameters
• 100,000 replicas of 𝜖 generated
• Sample sizes: n = 10 (very small) and n = 20 (common in applications)
• Probabilities of interest: between 10−1 and 10−3 (5
• Focus on autocovariance of order 1
Results for Cauchy Distribution (𝛼 = 1)
• a = 1: Approximation is extremely good for both n = 10 and n = 20
• a = 0.5: Approximation is excellent
• a = 0: Approximation is good enough for practical use, though not as precise as for
non-zero a
• a = -0.5:
– For n = 10: Accurate enough for practical use (critical values at levels < 3
– For n = 20: Accuracy decreases
• a = -1: Approximation not accurate in range of practical interest, especially as n
increases
Lower Tail Approximation for Negative a
• Theorem 11.2.2 provides approximation for 𝑃𝑛𝛾𝑛(1)≤ −𝑡
• Lower tail approximation is very sharp for a = -1 or a = -0.5
Student Distribution with 5 Degrees of Freedom (𝛼 = 5)
• For positive a: Approximation degenerates, becoming questionable for n = 10
• For a = 0: Approximation improves as 𝛼 increases
• For negative a: Approximation worsens as 𝛼 increases
General Observations
• For a = 0, approximation is fairly good until 𝛼 becomes large
• With many moments (large 𝛼), assuming normally distributed errors may give
better approximation
• Quality of approximation varies significantly with parameters a, n, and 𝛼
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