Course Information for STOR 755, Fall 2014
 
Estimation and Hypothesis Testing (Still under Construction)


 

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.

Office Hours: TBA
 

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 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.

    Exponential inequalities for sums of independent random variables via Chernoff bound: Hoeffding, Bennett, and Bernstein

    Azuma-Hoeffding inequality and McDiarmid's Bounded Difference inequality  

    The Glivenko-Cantelli theorem   

     Review of expectations and variances of random vectors 

    Definition and basic properties of the general multivariate Gaussian distribution

    Gaussian integration by parts, and applications

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

    Review of the multivariate central limit theorem and stochastic order symbols    

    The multivariate delta method and variance stabilizing transformations

    Consistency and asymptotic normality of method of moment estimators 

    Uniform convergence of convex functions

    Consistency and asymptotic normality of maximum likelihood estimators and empirical quantiles

    Extreme value theory for the Gaussian

    Bounds on the expected maximum of sub-Gaussian random variables

    Basics of the EM algorithm

    Convex functions and sets.  Gradient descent for convex functions.

    Talagrand's median concentration inequality for bounded convex functions

    Concentraion inequality for Lipshitz functions of Gaussian random variables

    Stein's method for normal approximation, with some applications. 

    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 text of Casella and Berger and 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 courseSubject to the instructor's approval, the paper can be one that is also related to the student's area of research.


Other sources:

``Concentraton 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)


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.