STA 7334: Assignments

AS = Asymptotic Statistics by A. W. van der Vaart
ELST = Elements of Large-Sample Theory by E. L. Lehmann.

Assignment 1 (Due Thurs, Sept 2)

AS: 2.15
ELST: 2.2.9, 2.3.12, 2.4.5(ii).

Assignment 2 (Due Thurs, Sept 16)

Prove that (ii) and (iv) are equivalent in the Portmanteau theorem.
AS: 2.10, 2.12
ELST:

Assignment 3 (Due Friday, Sept 24, by 1 p.m.)

Do this problem plus:
AS: 2.16, 2.17
ELST:

Assignment 4 (Due Friday, Oct 1, by 1 p.m.)

AS: 3.8 (but see note below)
ELST:
The second part of AS 3.8, concerning the expectation of 1/|Xbar|, can fail for n = 1, so you should prove the result for n >= 2. (Hint: Use convolution.) In addition, construct a counterexample to show that when n = 1, the expectation of 1/|Xbar| = 1/|X_1| may be finite under the conditions of the problem. (Hint: the density f does not have to be continuous at x = 0.)

Assignment 5 (Due Friday, Oct 29, by 1 p.m.)

Do this problem.

Assignment 6 (Due Friday, Nov 5, by 1 p.m.)

AS: 5.1, 5.2.

Assignment 7 (Due Friday, Dec 10, by Noon)

AS: 5.6, 5.19, 11.7, 11.10.

Additional Problems

ELST, Chapter 1

This is mostly review material from analysis. You should feel comfortable with most of these problems, but if not, try as many as you have time for.

AS, Chapter 2

You should be able to do all the problems at the end of Chapter 2. Most of them are straightforward. Some of the more interesting ones are 2.5, 2.8, 2.9, 2.10, 2.11, 2.12, 2.15, 2.16, 2.17, 2.18, 2.20, 2.21, 2.22, 2.23

ELST, Chapter 2

Again, you should be able to do most of these, but there are probably too many to try. Note that problem 2.2.1 is wrong, although it is supposed to be corrected in the 3rd printing.

AS, Chapter 3

3.2 was done in class. You should be able to do most of the rest, though some are more trouble than others. Definitely try 3.6 and 3.7. Simplifying the asymptotic variance in 3.4 might be a bit painful, but it should be relatively easy for the special case of normally distributed observations. Ditto for 3.5.

AS, Chapter 4.

4.1. 4.2 also looks interesting.

AS, Chapter 5.

5.3, 5.6, 5.7, 5.8, 5.9, 5.11, 5.12.

AS, Chapter 11.

11.5.