Homework Problems
Try to Submit them by the Next Class

  1. Let Z_{ij} be iid N(0,1).

    Let A = [ 1 3 2 0] B = [1 2] C = [1 2] [0 1 4 -2] |0 1] [3 4] [4 2] Let Y = AZB + C

    1. Find the mgf of Y.
    2. Identify the distribution of Y.
  2. Exercise 2 .
  3. Assigned Jan. 16, Number 10, page 324 of Arnold.
  4. Suppose W is Wishatp(n,Sigma) with Sigma pd. Determine E[tr(W-1)].
    (Use Lemma 17.10 p times.)
  5. Exercise 5 .
  6. Exercise 6 . Click here for the pilot data.
  7. Exercise 7 .
  8. Exercise 8 .
  9. Exercise 9: Prove that the estimator S for the linear model 

    Y = mu + e, mu \in V, V a p-dimensional subspace of R^n,
e is N_{n,r}(O, I, Sig)

    is an unbiased estimator of Sigma.

    DO NOT USE NORMALITY!!!

    Show can use e instead of Y, wlog. Then pcik off ithe (i,j)th expectation.
  10. Exercise 10: Your data for this exercise is in

    Just click on Click here for index then click on your data set

    Include the name of your data set in your report.

    This is a multivariate regression problem. The first four columns of your data set form the Y-matrix and the last 3 columns form the design matrix.

    1. Obtain the LS estimates of the regression coefficients.
    2. Obtain the estimate of Sigma.
    3. Record your results for beta-hat, associated standard errors, and error part of a 95\% CI in 3 matrices (side-by-side).
    4. Obtain a CI for beta_{11} - beta_{31}.
    5. Obtain a CI for beta_{11} - beta_{14}.
    6. Obtain the 4 residual plots.
  11. Click here for Exercise # 11
  12. Click here for Exercise # 12
  13. Click here for Exercise # 13
  14. Problem 5.4, p. 275 of Seber: Given x = (x_1,x_2), where x_1 and x_2 are random variables with zero means, unit variances, and correlation rho, find the proncipal components of x.
  15. Problem 5.7, p. 276 of Seber: Let x have mean 0 and disppersion matrix 

       [alpha + 1    1            1  ]
Sigma = [1         alpha +1       1   ]
   [1            1       alpha +1]

    Verify that the eigenvalus of Sigma - alphaI_3 are 3, 0, and 0. Hence, determine the eigenvalus of Sigma and then get the first pronciple component.

  16. First Part: Obtain the two tables discussed in class of the principal component analysis of the Test Scores Data Set (first 4 columns). Discuss the analysis. Name the principal components. (just use the R-function, pctable.r). Data is at Click for Data .
  17. Second Part of last problem: Obtain the normal q-q plots of the four principal components, i.e. the columns of Y = XT where the columns of T are the four eigenvectors. Next obtain the chi-squared q-q plot which was discussed in class on Tuesday. Comment on the plots.
  18. Click for the exercise on factor analysis.
  19. For this and the next two problems, if you don't have access to Seber wait until class Tuesday for a xerox copy of the problems. Number 6.2(a) in Seber, page 343.
  20. Number 6.4 (just the optimal rule, not the minimax) in Seber, page 343.
  21. Number 6.20 in Seber, page 346.
  22. Click for Exercise 22
  23. Suppose given Z that X1, ... , Xd are iid N(0,Z). Suppose also that n/Z has a Chi-squared distribution with n degrees of freedom. Prove that X = (X1, ... , Xd)' has a multivariate t-distribution with n degrees of freedom.
  24. Using the last problem generate some bivariate (d = 2) ts with degrees of freedom 1, 3, 10, 30. Generate them using only a normal generator. Use the language of your choice (in R use x = rnorm(n,0,sig) to generate n iid normals with mean 0 and standard deviation sig. Generate 200 or so of each so that the satterplot is informative. Obtain a 2 by 2 plot of the 4 scatterplots. Do two versions. One with just the scale the language chooses. Then do one with the same scale. In R this can be done with the following code (assuming that the t with degrees of freedom 1 is most variable). 

    par(mfrow=c(2,2))
plot(x1withdf1,x2withdf2,pch=" ",xlab="X1",ylab="X2")
points(x1,x2)
title(main="Whatever")

    Comment on the plots.

  25. For this problem, the data are in the file `genn2.dat.' This data file and the R functionts that you need are in this directory . This is a single sample data set with 4 dimensions. Use the LS, Sign, and Wilcoxon analyses to anaylyze this data by obtaining for each analysis:
    1. Test that all means are 0.
    2. Estimates, SEs, and CIs for all four location parameters. Check to see if the CIs caught the true mean, (4,3,2,1).
    3. Normal q-q plots for each component.
    4. Discuss and compare the analyses.
  26. This problem is to verify the validity (actually the code I wrote) of the three analyses above for elliptical normal errors. Let X be multivariate normal with dimension 4 and variance I. The variable of interest is Y = CX where C is 

         16.475762  2.322324  7.870976 -2.252274
C=    2.322324 15.880716  3.023949 -6.226766
      7.870976  3.023949 23.169431 -3.987143
     -2.252274 -6.226766 -3.987143 18.574489
    Hence the null hypothesis is true (i.e., mu = 0). Set the sample size at 25, i.e 25 iid observations of Y, i.e., each data matrix is 25 by 4. Simulate 1000 such samples. Collect the test statistics and p-values of the LS, Sign, and Wilcoxon analyses. Use nominal alpha = .05.
    1. Report the empirical alpha levels.
    2. Obtain uniform q-q plots of the empirical alpha levels. Recall that under the null, p-values, hopefully, have a uniform distribution.