Estimation and Hypothesis Testing

**Class
meetings: **Monday and Wednesday 2-3:15pm in
Hanes 125.

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

**Registration:
**Enrollment and registration for the course
is handled online. ** **

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

**Office:
**Hanes 308 **Phone: **919-962-1352.

**Office
Hours: **TBA

Course Overview:
STOR 755 is a PhD level course covering advanced
topics in statistical inference. The course is
appropriate for graduate students in Statistics,
Biostatistics and related fields of study. The
focus of the course is theoretical, emphasizing both
general results and the mathematical techniques
underlying their proof. Topics will be presented
in a detailed, mathematically rigorous fashion, and will
be as self-contained as the material allows. We
will cover asymptotic and non-asmptotic results,
with an emphasis on non-parametric problems.

**
Texts: **There is no primary text for the
course. Some of the initial material is covered in
the book "Asymptotic Statistics" by Aad van der Vaart
and the book. Other material will be drawn from
tutorials, surveys, and research monographs.

**
Tentative Syllabus: **The course
will cover a broad range of topics, covering theoretical
tools of importance and broad applicability in modern
theoretical statistics. We will emphasize results
whose proofs illustrate techniques of general
importance. The following is a unordered,
tentative list of topics. The final list will
depend on available time and, to some extent, student
preferences.

The Glivenko-Cantelli
theorem

The multivariate
delta method

Sample percentiles:
consistency and CLTs via the Bahadur representation

Consistency and asymptotic normality of M and Z estimators

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

Basic concentration
inequalities for bounded and Gaussian random variables

Interpolation and
comparison theorems for Gaussian random variables

Stein's method for normal
approximation. Berry-Esseen theorems

The Vapnik-Chervonenkis
inequality and the VC dimension

Overview of some basic
results concerning chaining and empirical process
theory

**Prerequisites:**
Students should look over Chapter 2 of van der Vaart for
a review of prerequisite material for the course.
Additional material can be found in Chapter 1 of

**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.

"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.

"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.