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.

**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
chapters 3-5 of the book "Asymptotic Statistics" by Aad
van der Vaart. Other material will be drawn from
tutorials, surveys, and research monographs.

**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.

Azuma-Hoeffding
inequality and McDiarmid's Bounded Difference inequality

The Glivenko-Cantelli theorem

Gaussian
integration by parts, and applications

General comparison
theorem for Gaussian random vectors; Slepian's lemma

The multivariate delta method and variance stabilizing transformations

Consistency
and asymptotic normality of method of moment estimators

Uniform
convergence of convex functions on compact sets

Asymptotic
normality of maximum likelihood estimators and empirical
quantiles

Some extreme value theory

Bounds on
the expected maximum of sub-Gaussian random variables

Talagrand's
median concentration inequality for bounded convex
functions

Concentraion inequality for Lipshitz functions of Gaussian
random variables

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 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:**

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

``Topics in Random Matrix
Theory", by Terrence Tau (Concise treatment of some
basic concentration results.)

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

**Background reading**

``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)

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