**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 **F**m,
**G**n are different
from the empirical-process usage of the chapter: here they are
"empirical distribution functions". That is, **F**m(t) is the proportion of
observations X1, ...,
Xm less than or equal to t,
and **G**n(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.