Class meetings: Tuesday and Thursday 2-3:15pm, in HANES 107.
Prerequisites: Working knowledge of theoretical statistics and measure theoretic probability at the introductory graduate level. Some real analysis.
Registration: Enrollment and registration for the course is handled by Charlotte Rogers in the Department Statistics and Operations Research. Ms. Rogers can be reached at 962.2307, or by email at firstname.lastname@example.org.
Instructor: Andrew B. Nobel, Department of Statistics and Operations Research
Office: Hanes 308 Phone: 962-1352.
Hours: TBA and by appointment.
Overview: STOR 755 is a PhD level course in
advanced statistical inference that is appropriate for graduate
Statistics, Biostatistics and related fields. The focus of the
course is on theory, both results and techniques, rather than specific
applications. Topics will be presented in a detailed,
rigorous fashion, and will be as self-contained as the material
The primary emphasis of the course will be on asymptotic results in
Texts: The primary text for the course is "Asymptotic Statistics" by Aad van der Vaart. We will also cover some material from ``Mathematical Statistics'' by Jun Shao.
Syllabus: The course will cover selected topics from the texts above, as well as material from other sources, with an emphasis on results whose proofs illustrate techniques of general importance. The following is a tentative list of topics. The final list will depend on available time and, to some extent, student preferences.
McDiarmid's bounded difference inequality
Empirical CDFs and the Glivenko-Cantelli
The Vapnik-Chervonenkis inequality
Weak version of the Dvoretsky, Kiefer and Wolfowitz inequality
Sample percentiles: consistency and CLTs
via the Bahadur representation
The multivariate delta method
Moment estimators: Existence and CLTs via
the inverse function theorem
M and MLEs: consistency
and asymptotic normality
Projections onto sums
U-statistics (one and two sample): asymptotic normality
Rank tests: asymptotic normality
Consistency and relative efficiency of
Overview of some basic results concerning
chaining, weak convergence of processes and empirical process
theory. Improved Vapnik Chervonenkis inequalities.
Most of the material is
from Chapters 3, 4, 5, 11, 12, 13, 14, and 22 of van der Vaart, and
from parts of Chapters 5 and 6 of Shao. The initial material on
McDiarmid's inequality and the Vapnik-Chervonenkis inequality can be
found in the first three chapters of the book of Devroye and Lugosi
below. The final section concerning empirical process
theory will follow van der Vaart, chapters 18-20 and select parts of
van de Geer.
Chapter 1 of Shao and Chapter 2
of van der
Vaart cover most of the prerequisite material needed for the
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
"Statistical Inference", Second Edition, by G. Casella and R.L. Berger, Duxbury, 2002.
Statistics", Second Edition, by P.J. Bickel and K.A.
Doksum, Prentice Hall, 2001.
Methods in Density Estimation", by L. Devroye and G.
Lugosi, Springer, 2001. (See
also the nice lecture notes on concentration inequalities by Boucheron,
Lugosi and Massart, available from Lugosi's web page.)
and Measure", Third Edition, by P. Billingsley, Wiley,
"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. (A very well written and insightful book on basic inequalities, which can be read with relatively little background. Contains lots of nice problems, with solutions.)
"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.
are expected to adhere to the UNC honor code at all