Math 439 Homework 3

Math 439 Homework 3 - Fall 2008

HOMEWORK #2 due 10-14

In the following, _ means subscript and ^ means superscript.

NOTE: In all problem sets that use SAS, arrange your answers into three parts, in the following order:

(I) Your answers to all questions, either written by hand or using a word processor,

(II) The SAS program source files (*.sas files) that you used in the problem set,

(III) The output from the SAS programs in Part II.

In Part I, you can refer to plots or tables or large matrices that problems ask for by saying (for example), ``The scatterplot or matrix for Problem 3 is on page 17 of the SAS output.'' If necessary, add page numbers to the SAS output, so that (for example) you don't have several different page 1s in Part III.

1. Assume that X is vector-valued normal N(mu_X,B) for

            (  5 )                 ( 0   0   0 )
     mu_X = ( -3 )    and    B  =  ( 0   2   3 )
            ( -2 )                 ( 0   3   5 )

Consider the random vector Y = A X in R^2 for

     A  =  ( 1   2   3 )
           ( 0   1   2 )

Thus Y is normal N(mu_Y,C_Y) for some vector mu_Y and matrix C_Y. Find mu_Y and C_Y.

2. Let X be an n by d matrix whose entries X_ia are independent N(0,1) for 1 le i le n and 1 le a le d.

(i) Let Y be the random matrix X'X. Show that Y has a Wishart distribution W_d(n,I_d), in the notation of equation (4.15) in the text. (Hint: Write out the entries of Y = (Y_ij) or Y = (Y_ab) in terms of the entries X_{ia} of X. Note that it is sufficient to set mu=0 in (4.15) in the text.)

(ii) Show that E(Y)=n I_d.

3. Table 5.5 (page 150) in the text has four measurements on m=19 beetles from the flea beetle species Haltica oleracea and from n=20 specimens of another flea-beetle species, H. carduorum. (See also the data file FleaBeetles.dat.)

(i) Using the Hotelling T^2 test for all four measurements y_1,y_2,y_3,y_4, use SAS to test the hypothesis H_0:E(X)=E(W), where X_1,...,X_m (in R^4) represent the measurements from H. oleracea and W_1,...,W_n the measurements from the second flea-beetle species. Do you accept or reject H_0?

(ii) From the output, what is the value of the associated F statistic for the multivariate test? What is the number of degrees of freedom, both in the numerator and in the denominator? How were these derived from the number of components in the observations (d=4) and the sample sizes (m,n)?

(iii) Carry out two-sample t-tests on the four measurements y_1,y_2,y_3,y_4 individually. Which of these are significantly different between the two samples? What are the two-sided P-values?

(Hints: See MLizards.sas on the Math439 Web site. Do not log-transform the data. If you use proc format to assign descriptive tags to the Species variable (=1,2), make sure that you use the correct species names.)

(Warning: Make sure that SAS reports that you have measurements for 39 individuals.)

4. Consider the blood glucose data in Table 3.8 (page 80) of the text or MPairedSamp.sas on the Math439 Web site. Let y_1,y_2,y_3 be the fasting blood levels of each subject and x_1,x_2,x_3 the levels after one hour, as in Table 3.8. Let z_i=x_i-y_i (i=1,2,3) be the increases (as in MPairedSamp).

(i) Use SAS to test H_0:E(z_1)=E(z_2)=E(z_3); that is, that the increase in blood glucose levels was the same for all three visits across the subjects. Do you accept or reject H_0?

(Hint: Note that a=b=c is equivalent to b-a=0 and c-a=0. Thus, you can define W1=Z2-Z1 and W2=Z3-Z1 and then test H_0:E(W)=0 for W = (W_1,W_2)'. See also the discussion in Section 5.9.1 in the text.)

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