2 hour statsitcis test on multivariate analysis on 10 nov

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Question 1. [19 marks]

Suppose the random variables X1, X2 have the covariance matrix

� = � 1 −2−2 5 �

with the eigenvalues-eigenvectors pairs given as follows

λ1 = 0.172, e1 = (−0.924, −0.383)T, λ2 = 5.828, e2 = (−0.383, 0.924)T,

(a) [4 marks] Explain how to generate random vectors from the multivariate normal distribution

�(�, Σ), where � = (0, 0)� and Σ is given above.

(b) [4 marks] Using the following four N(0, 1) random numbers to generate 2 random vectors (each has the dimension of 2 × 1): 0.86, 0.67, − 0.18, 0.25.

(c) [2 marks] Use the above eigenvalues and eigenvectors, write down the singular value

decomposition expression for Σ.

(d) [3 marks] Calculate the factor loadings if we use only the second eigen value and vector (i.e. �� and ��) to perform Factor Analysis.

(e) [3 marks] Hence calculate the corresponding specific variances for the two random variables.

(f) [3 marks] Express the random variables ��, �� using the factor loadings derived above.

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Question 2. [26 marks]

(a) [6 marks] Consider a -dimensional random vector � where �~�"(0"×�, Σ"×"). Answer the follow questions.

(i) What is the distribution of �� − 2��, where �� and �� are both from the above normal distribution? (You need to specify the distribution, its mean and covariance matrix.)

(ii) Briefly explain why �� Σ#�� follows $�( ) distribution. (iii) Suppose %"×" is a symmetric and positive definite matrix and therefore there exists a “square-root” matrix %�/� such that % = %�/�%�/� . Find the distribution of %�/��.

(b) [5 marks] Let ��, … , �( be a multivariate random sample, where each �) has the dimension of × 1. (i) Explain how to check if a random sample ��, … , �( follow a multivariate normal distribution. (ii) If ��, … , �( fail the normality checking, discuss how to transform each component of these random vectors to univariate normal random variables.

(c) [4 marks] Consider two random variables (��, ��) (i) If �� follows a normal distribution and �� is also normally distributed, will (��, ��) jointly follow a bi-variate normal distribution? (No need to give a reason.)

(ii) If (��, ��) follow a bi-variate normal distribution, will �� or �� are also normally distributed? Why? (In this part, you need to provide reasons briefly when answering “why”.)

(d) [5 marks] Suppose 15 multivariate observations are obtained on 3 variables and these observations are denoted by *�, … , *�+, where each *) is a 3 × 1 vector containing 3 values from the 3 variables. Consider the population mean vector � = (��, ��, �,)� . Suppose we wish to test the following hypotheses involving different contrasts:

-.: �� − �� + �,

2 = 0, �� − �� + �,

2 = 0, �, − �� + ��

2 = 0,

and significance level for the test is 1 = 5%. Clearly explain how to test this hypothesis. In your answer you must include: (i) assumptions made, (ii) test statistic, and (iii) how to decide if to

reject or retain the null hypothesis.

(e) [6 marks] This question considers the data and contrasts specified in part (d) above.

Simultaneous confidence intervals are often needed, particularly when the null hypothesis is

rejected. Answer the following questions (with the overall confidence level of 95%).

(i) Write down the T2 simultaneous confidence intervals for all the contrasts. You are required

to simplify the answers to as much as you can.

(ii) Write down the Bonferroni simultaneous confidence intervals for all the contrasts. You are

required to simplify the answers to as much as you can.

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Question 3. [6 marks]

(a) [3 marks] Briefly explain the weighted least-squares method to estimate factor scores in factor

analysis.

(b) [3 marks] Briefly explain how to construct a Q-Q plot.

Question 4. [12 marks]

(a) [5 marks] In order to assess the effectiveness of a particular drug to stabilize blood pressure, a

multivariate random sample of size 30 is obtained from 30 patients. From each patient, whose blood pressure measure is taken across 14 consecutive days, and these measures form a 14-

element vector for blood pressures. For individual 3, the blood pressure vector is denoted by *) = 4*)�, … , *),�56

� . The hypothesis we wish to test is

-.: �� = �� = ⋯ = ��5

where �< represents mean blood pressure on day = after taking the drug. Clearly explain how to perform this test. Develop your answer as far as you can. In your answer, you need to include:

assumptions made, test statistic and how do you decide to reject or retain the null hypothesis.

Use 1 = 5% significance level.

(b) [7 marks] Now suppose the doctor suspects this medicine could affect males and females

differently for controlling blood pressure. Answer the following questions assuming the data are

still the same as in part (a). Assume there are 20 males and 10 females in the sample.

(i) Write down the null hypothesis that can be used to test the doctor’s suspicion.

(ii) Explain how to test if the populations of Male and Female have equal covariance matrices.

(iii) Then Provide a detailed procedure to test the hypothesis specified in (i) under the

assumption of equal covariance matrices. You must give details of the test, including

assumptions, the test statistic, distribution of the test statistic and the hypothesis testing

decision rule.

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Question 5. [12 marks]

Consider the following multiple linear regression model:

>) = ?. + ?�*�) + ?�*�) + ?,*,) + @)

where @) iid �(0, C�) and 3 = 1, … , 25.

(a) [5 marks] Now suppose we wish to test -.: ?� = 1 and ?� = 0. Clearly explain how to perform this test at 1 = 95% level. Develop your answer as far as you can. In your answer, you need to include: test statistic and how do you decide to reject or retain the null hypothesis.

(b) [7 marks] Explain how to construct Bonferroni simultaneous 95% CI for contrasts ?� − ?�, ?� − ?, and ?� − ?,.