Testing Hypothesis for Autoregressive Coefficient in
Autoregressive Processes
Alternative Approximation Techniques
• When our approximation is poor, consider:
– Edgeworth expansion (poor in relative error)
– Approximating Student distribution with normal distribution
• Caution: For small 𝛼, the "right" variance may not match the Student distribution
• Normal approximation with calibrated variance doesn’t consistently improve
results
• Note: Symmetry in Student distribution reduces approximation precision
• More work needed for comprehensive approximations across various regimes
Testing Hypothesis for Autoregressive Coefficient
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
1. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
2. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
3. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
4. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
5. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
6. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
7. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
8. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
9. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
10. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
11. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
12. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
13. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
14. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
15. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
16. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
17. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
18. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
19. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
20. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
21. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
22. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
23. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
24. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
25. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
26. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
27. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
28. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
29. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
30. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
31. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
32. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
33. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
34. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
35. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
36. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
37. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
38. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
39. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
40. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
41. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
42. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
43. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
44. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
45. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
46. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
47. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
48. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
49. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
50. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
51. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
52. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
53. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
54. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
55. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
56. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
57. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
58. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
59. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
60. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
61. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
62. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
63. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
64. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
65. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
66. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
67. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
68. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
69. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
70. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
71. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
72. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
73. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
74. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
75. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
76. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
77. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
78. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
79. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
80. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
81. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
82. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
83. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
84. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
85. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
86. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
87. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
88. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
89. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
90. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
91. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
92. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
93. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
94. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
95. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
96. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
97. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
98. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
99. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
100. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
101. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
102. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
103. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
104. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
105. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
106. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
107. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
108. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
109. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
110. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
111. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
112. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
113. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
114. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
115. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
116. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
117. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
118. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
119. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
120. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
121. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
122. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
123. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
124. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
125. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
126. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
127. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
128. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
129. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
130. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
131. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
132. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
133. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
134. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
135. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
136. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
137. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
138. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
139. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
140. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
141. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
142. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
143. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
144. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
145. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
146. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
147. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
148. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
149. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
150. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
151. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
152. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
153. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
154. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
155. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
156. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
157. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
158. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
159. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
160. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
161. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
162. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
163. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
164. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
165. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
166. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
167. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
168. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
169. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
170. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
171. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
172. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
173. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
174. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
175. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
176. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
177. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
178. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
179. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
180. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
181. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
182. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
183. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
184. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
185. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
186. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
187. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
188. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
189. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
190. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
191. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
192. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
193. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
194. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
195. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
196. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
197. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
198. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
199. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
200. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
201. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
202. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
203. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
204. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
205. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
206. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
207. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
208. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
209. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
210. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
211. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
212. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
213. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
• Classical estimation: 𝑎=𝛾𝑛(1)/𝛾𝑛(0)
• Test problem: 𝐻0:𝑎≤𝑎0 vs 𝐻1:𝑎>𝑎0
• Test statistic: 𝛾𝑛(1)−𝑎0𝛾𝑛(0)
• Important: Use 𝛾𝑛(0)=𝑛−1∑𝑋𝑖2
1≤𝑖≤𝑛−1 instead of 𝛾𝑛(0)
Tail Approximation for Test Statistic
Theorem 11.2.2: For an AR(1) process with Student-like innovations (parameter 𝛼):
214. If 𝑎>𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼𝛼𝛼−1
22(𝑎−𝑎0)𝛼/2 ∑ (1−𝑎2(𝑛−𝑖)
1−𝑎2)𝛼/2
1≤𝑖≤𝑛−1 𝑡−𝛼/2
215. If 𝑎0=𝑎≥0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝐾𝑠,𝛼
2𝛼𝛼2 ∑ (𝑛−𝑘)
1≤𝑘≤𝑛−2 𝑎(𝑘−1)𝛼𝑡−𝛼log𝑡
216. If 𝑎0>0 and 𝑎<𝑎0:
𝑃{𝑛(𝛾𝑛(1)−𝑎0𝛾𝑛(0))>𝑡}∼𝑐(𝑎0,𝑎,𝛼,𝑛)𝑡−𝛼
Implications for Hypothesis Testing
• Different tail probability decays based on 𝑎 and 𝑎0
• Recommendation: Use asymmetric confidence intervals
• For small first type risk 𝜂, critical value: 𝑡𝜂=(𝑐(𝑎)/𝜂)2/𝛼
• Actual risk may be higher than intended, especially for 𝑎 close to 1
Autoregressive Processes of Arbitrary Order
Theorem 11.3.1: For AR(p) process with Student-like innovations:
• There exist semialgebraic sets 𝑅𝑘 in ℝ𝑝 and a function 𝑐(𝜃) such that:
𝑃{𝑛𝛾𝑛(1)>𝑡}∼
{
𝑐(𝜃)𝑡−𝛼/2 if 𝜃∈ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
𝑐(𝜃)𝑡−𝛼log𝑡if 𝜃∉ ⋃𝑅𝑘
1≤𝑘≤𝑛−1
• Key points:
– Only two possible tail behaviors: 𝑡−𝛼/2 or 𝑡−𝛼log𝑡
– Regions 𝑅𝑘 are nested as 𝑛 increases
– Regions depend on 𝑝 but not on 𝑛
