Estimation and Hypothesis Testing

**Class
meetings: **Tuesday
and Thursday 2-3:15pm, in HANES 107.

**Prerequisites:
**Working
knowledge of theoretical statistics and measure theoretic probability
at the introductory graduate level. **
**Some real analysis.** **

**Registration:
**Enrollment
and registration for the course is handled by
Charlotte Rogers in the Department Statistics and Operations Research. Ms. Rogers can be reached at 962.2307,
or by email at crogers@email.unc.edu. **
**

**Instructor:
**Andrew
B. Nobel, Department of Statistics and Operations
Research

**Office:
**Hanes
308 **Phone: **962-1352.

**Office
Hours: **TBA and by appointment.

** **

Course
Overview: STOR 755 is a PhD level course in
advanced statistical inference that is appropriate for graduate
students in
Statistics, Biostatistics and related fields. The focus of the
course is on theory, both results and techniques, rather than specific
applications. Topics will be presented in a detailed,
mathematically
rigorous fashion, and will be as self-contained as the material
allows.
The primary emphasis of the course will be on asymptotic results in
non-parametric settings.

**
Texts:
**The primary text for the course is "Asymptotic
Statistics"
by Aad van der Vaart. We will also cover some material from

**
Syllabus:
**The
course will cover selected topics from the texts above, as
well as material from other sources, with an emphasis
on results whose proofs illustrate techniques of general
importance. The
following is a tentative list of topics. The final list will
depend on available time and, to some extent, student preferences.

McDiarmid's bounded difference inequality

Empirical CDFs and the Glivenko-Cantelli
theorem

The Vapnik-Chervonenkis inequality

Weak version of the Dvoretsky, Kiefer and
Wolfowitz inequality

Sample percentiles: consistency and CLTs
via the Bahadur representation

The multivariate delta method

Moment estimators: Existence and CLTs via
the inverse function theorem

M and MLEs: consistency
and asymptotic normality

Projections onto sums

U-statistics (one and two sample):
asymptotic normality

Rank tests: asymptotic normality

Consistency and relative efficiency of
tests

Overview of some basic results concerning
chaining, weak convergence of processes and empirical process
theory. Improved Vapnik Chervonenkis inequalities.

Most of the material is
from Chapters 3, 4, 5, 11, 12, 13, 14, and 22 of van der Vaart, and
from parts of Chapters 5 and 6 of Shao. The initial material on
McDiarmid's inequality and the Vapnik-Chervonenkis inequality can be
found in the first three chapters of the book of Devroye and Lugosi
below. The final section concerning empirical process
theory will follow van der Vaart, chapters 18-20 and select parts of
van de Geer.

Chapter 1 of Shao and Chapter 2
of van der
Vaart cover most of the prerequisite material needed for the
course.

**Grading:
**Reading assignments and homework problems will be set periodically
throughout the semester. In addition, students will be asked to
read and offer written or oral comments on a paper in the
literature. More details on the project will be provided later in
the semester.

Other sources:

"Statistical Inference", Second Edition, by G. Casella and R.L. Berger, Duxbury, 2002.

"Mathematical
Statistics", Second Edition, by P.J. Bickel and K.A.
Doksum, Prentice Hall, 2001.

"Combinatorial
Methods in Density Estimation", by L. Devroye and G.
Lugosi, Springer, 2001. (See
also the nice lecture notes on concentration inequalities by Boucheron,
Lugosi and Massart, available from Lugosi's web page.)

"Probability
and Measure", Third Edition, by P. Billingsley, Wiley,
1995.

"Multivariate Analysis", by K.V. Mardia, J.T. Kent and J.M. Bibby,
Academic Press, 1979.

"The Cauchy-Schwartz Master Class'', J.M. Steele, Cambridge,
2004. (A very well written and insightful book on basic
inequalities, which can be read with relatively little
background. Contains lots of nice problems, with solutions.)

"Principles of Mathematical Analysis", Third Edition, by W. Rudin,
McGraw Hill. (This book is a good reference for real analysis.)

Jon Wellner at U.W. Seattle has extensive lecture notes on theoretical
statistics available on his web page.

**Honor
Code: **Students
are expected to adhere to the UNC honor code at all
times.