3. Let X = (X1, X2, X3, X4)T be a multivariate normal random vector with mean u and variance E. Suppose that a random sa

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3 Let X X1 X2 X3 X4 T Be A Multivariate Normal Random Vector With Mean U And Variance E Suppose That A Random Sa 1 (200.42 KiB) Viewed 36 times

3. Let X = (X1, X2, X3, X4)T be a multivariate normal random vector with mean u and variance E. Suppose that a random sample of size 20 from the same distribution as X is observed, and the observed data are recorded in the file "dat.csv". In the following sub- questions, you can use R or any computer software to compute parameter estimates or perform matrix operations, but you need to show the steps in your derivations. (a) Compute the maximum likelihood estimates (MLE) of u and £. (b) Using the MLE, estimate the multiple correlation coefficient between X, and (X2, X3). (c) Using the MLE, estimate the multiple correlation coefficient between X and (X2, X3, X4) (d) Using the MLE, estimate the partial correlation coefficient between X, and X2, ad- justing for X3 and X4. (e) Using the MLE, estimate the variance and the loadings of the first principal component. Construct a 95% confidence interval for the variance of the first principal component.