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
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.
inequality and McDiarmid's Bounded Difference inequality
The Glivenko-Cantelli theorem
integration by parts, and applications
theorem for Gaussian random vectors; Slepian's lemma
The multivariate delta method and variance stabilizing transformations
and asymptotic normality of method of moment estimators
convergence of convex functions on compact sets
normality of maximum likelihood estimators and empirical
Some extreme value theory
the expected maximum of sub-Gaussian random variables
median concentration inequality for bounded convex
Concentraion inequality for Lipshitz functions of Gaussian
and packing numbers for pseudo-matric spaces
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
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.
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.
Inequalities: A Nonasymptotic Theory of Independence", by
S. Boucheron, G. Lugosi, and P. Massart (An
extensive treatment of concentration inequalities)
``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)
Second Edition, by G. Casella and R.L. Berger, Duxbury,
2002. (Introductory graduate level text on
``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.