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 tentative list of
topics: material covered will depend on available time
and, to some extent, student preferences.

Overview of exponential inequalities for sums of independent
random variables: Chernoff bound, and inequalities
of Hoeffding, Bennett, and Bernstein

The Glivenko-Cantelli theorem

Expectations
and variances of random
vectors.

Definition and basic properties of the
general multivariate Gaussian distribution.

Review of background material, including the multivariate central limit theorem, O_P and o_P notation

The multivariate delta method and applications, including variance stabilizing transformations

Consistency and
asymptotic normality of method of moment
estimators

Consistency and asymptotic normality of maximum
likelihood estimators and empirical quantiles

Extreme value theory (Gumbel limit) for the Gaussian

Basics of the EM algorithm

Convex functions and sets. Gradient descent
for convex functions.

Basic concentration
inequalities for bounded and Gaussian random variables

Elementary interpolation ideas and comparison theorems
for Gaussian random variables

Stein's method for normal
approximation, with some
applications.

Covering and packing numbers for pseudo-matric
spaces

Lower and upper bounds for the expected maximum of a
Gaussian process: minoration
and the Dudley integral

Vapnik-Chervonenkis theory of uniform laws of large
numbers, including shatter coefficients and
combinatorial dimension

Basics of empirical process theory

**
Prerequisites:** Students should look over text of
Casella and Berger and Chapter 2 of van der Vaart for a
review of prerequisite material 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 prepare a brief written summary of a
research/survey/tutorial paper
related to the material in the
course. Subject
to the instructor's approval, the paper can be one
that is also related to the student's area of
research.

Other sources:

"Statistical
Inference", Second Edition, by G. Casella and R.L.
Berger, Duxbury, 2002. (Introductory graduate
level text on theoretical
statistics)

"Mathematical Statistics", Second Edition, by P.J. Bickel and K.A. Doksum, Prentice Hall, 2001. (Introductory graduate level text on theoretical statistics)

"Combinatorial
Methods in Density Estimation", by L. Devroye and G.
Lugosi, Springer, 2001. (Covers some elementary concentration
inequalities, and VC-theory)

"Multivariate Analysis", by K.V. Mardia, J.T. Kent and
J.M. Bibby, Academic Press, 1979. (Graduate text
on multivariate analysis)

"The Cauchy-Schwartz Master Class'', J.M. Steele,
Cambridge, 2004. (Well-written book covering basic
inequalities.)

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.