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.
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
and variances of random
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
asymptotic normality of method of moment
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.
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
Covering and packing numbers for pseudo-matric
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
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.
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
related to the material in the
to the instructor's approval, the paper can be one
that is also related to the student's area of
Inference", Second Edition, by G. Casella and R.L.
Berger, Duxbury, 2002. (Introductory graduate
level text on theoretical
"Mathematical Statistics", Second Edition, by P.J. Bickel and K.A. Doksum, Prentice Hall, 2001. (Introductory graduate level text on theoretical statistics)
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.
Code: Students are expected to adhere to the UNC
honor code at all times.