Homework Assignments, Stat 710, Fall 2007

Homework Set #1, due Monday September 17 : in Chapter 5 of van der Vaart text, #5.12, 5.14, 5.18, 5.25. As a final assigned probem, prove the assertion on p.46, lines 3-6 from the bottom, "One simple set of sufficient conditions [for {m&theta(.)}&theta to be a Glivenko-Cantelli class] is ...". (The hint, as we shall discuss in class, is to study the proof of Theorem 5.14.)   For HW1 partial solutions and comments, click here.

Homework Set #2, due Friday October 5 : in Chapter 19 of van der Vaart text, #19.3, 19.4, 19.5, 19.6, 19.7, and 19.10.   Notes. Problem 19.3 involves only checking equality of covariances and invoking an appropriate Theorem to imply that a unique set of finite-dimensional distributions determines a unique stochastic process law. In problem 19.4, the meaning of the notations Fm, Gn are different from the empirical-process usage of the chapter: here they are "empirical distribution functions". That is, Fm(t) is the proportion of observations X1, ..., Xm less than or equal to t, and Gn(t) is the proportion of observations Y1, ..., Yn less than or equal to t. Problem 19.4(c) and 19.10 are exercises in formulating limiting probabilities using empirical process convergence plus continuous mapping Theorem. Problem 19.5 is about bracketing and is fairly straightforward. 19.6 and 19.7 give some practice in estimating the VC numbers used to measure the size of function classes used in proving GC and Donsker properties.   For HW2 solutions, click here.

Homework Set #3, due Monday October 29 : Chap. 6, p.91: # 1, 2, 3, 4, 6.
For HW3 solutions, click here.

Homework Set #4, due Friday November 16: Chap. 7, p.106, #1, 5, 6, 10. Chap. 8, p.123, # 3.
For HW4 solutions, click here.

Homework Set #5, due Wednesday, Dec. 12. There are 7 problems in all. The reading for this part of the course is from Notes of mine, about Martingale Methods, together with a little bit of material on Chapter 13 (from which some problems will be taken). This part of the course is about compensated counting processes and martingale methods in statistics, especially with reference to rank-based statistics.
      (1). Prove that when Xi   for   i=1,2,...,n,    are iid with not necessarily continuous distribution function F, then the Kiefer-Wolfowitz Generalized Nonparametric Maximum Likelihood Estimator of the unknown d.f. F based on the sample of size n is the empirical distribution function. (For this problem, you can briefly summarize the argument given in class to establish that the GNPMLE must assign all its mass to probability atoms at the points Xi.)
      (2). Suppose that (Xi, Zi) are iid where Xi given Zi has continuous cumulative hazard function exp(&beta' Zi) &Lambda(t). Then we saw in class that the Cox Partial Likelihood is equal to the product of Likelihood factors involving (&beta,&Lambda) with &Lambda replaced by the weighted Nelson-Aalen estimator &Lambda&beta (found in class). Show that this same Cox Partial Likelihood expression is equal to the "marginal rank likelihood", that is, to the probability conditional on the Z's that the Xi observations would fall in their observed sorted order.
      (3)--(4). Two problems from Chapter 13, numbers 4 and 6 on pp.190-191. But for #13.6, you may if you prefer do the problem for the 2-sample (not the signed-rank) version of the Wilcoxon.
      (5)--(7). The remaining problems are about the martingale material: Exercises 2 and 3, respectively on pp. 8 and 11 of Chapter 1 of the Martingale Methods in Statistics notes referenced as Handout (3) below, and Exercise 5 on p. 44 of those Notes. In the latter problem find also the variance process of the process   M(t).

© Eric V Slud, December 5, 2007.