This special topics course will treat modern empirical process theory and applications of this theory to a variety of statistical problems. Applications will be drawn from survival analysis, econometrics, semiparametric models, regression and density function estimation, clustering, and classification. I plan to draw on several recent books, including those by R. M. Dudley and S. van de Geer. During the last weeks of the quarter I will cover some of the new isoperimetric inequalities and applications of these inequalities to problems in model selection and adaptive nonparametric estimation.

Homework: Assignments will cover complements to the theory, and introduce students to the basics of empirical process reasoning and methods. Course grades will be based on the homeworks and class participation for those taking the class for credit.

Prerequisites: Statistical theory at the level of Statistics 582, and some familiarity with probability theory, ideally at the level of Statistics 521.

Textbooks:

• Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York
• Van de Geer, S. (2000). Empirical processes in M-estimation. Cambridge University Press.
• Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge University Press.

  
  

  Tentative Topics / Outline
  • Empirical Process Basics:
    Exponential bounds and Chaining;
    Empirical Processes and Gaussian Limits;
    Vapnik - Chervonenkis classes of sets and functions;
    Uniform covering numbers; bracketing covering numbers;
    Glivenko-Cantelli classes; Donsker classes;
  • Applications of Empirical Processes to Statistics:
    Likelihood estimation in semiparametric models;
    nonparametric maximum likelihood; regression.
  • Adaptive Nonparametric Estimation: isoperimentric inequalities and model selection
    isoperimetric inequalities;
    model selection.

  Books:

  • Van der Vaart, A. W. and Wellner, Jon A. (1996). Weak Convergence and Empirical Processes, Springer, New York.
  • Van de Geer, S. A. (1999). Applications of Empirical Process Theory Cambridge University Press, Cambridge.