Course Information for STOR 755 (Spring 2009)
 
Estimation and Hypothesis Testing

 

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 crogers@email.unc.edu.  

Instructor:  Andrew B. Nobel, Department of Statistics and Operations Research

Office: Hanes 308   Phone: 962-1352.

Office Hours: TBA and by appointment.
 

Course Overview: STOR 755 is a PhD level course in advanced statistical inference that is appropriate for graduate students in 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, mathematically rigorous fashion, and will be as self-contained as the material allows.  The primary emphasis of the course will be on asymptotic results in non-parametric settings. 


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 theorem

    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 tests

    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 course. 


Grading: 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 the semester.


Other sources:

"Statistical Inference", Second Edition, by G. Casella and R.L. Berger, Duxbury, 2002.

"Mathematical Statistics", Second Edition, by P.J. Bickel and K.A. Doksum, Prentice Hall, 2001.

"Combinatorial 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.)

"Probability and Measure", Third Edition, by P. Billingsley, Wiley, 1995.

"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.
 

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