Shape Restrictions, Empirical Processes, and Semiparametric Models

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Empirical process theory, which originated in the fundamental question of how to estimate a probability distribution from sample data, has developed into a powerful set of tools and unifying methods for the study of the properties of a wide range of statistical procedures. This research project aims to broaden and deepen these tools and to develop additional methods. It is anticipated that this theoretical work will enable progress in several application areas, including sampling designs used in epidemiology and clinical trials, as well as models used for HIV-AIDS data. The project also involves training of graduate students and development of graduate level courses. The investigator plans to study new statistical methods for shape-restricted models in univariate and multivariate settings. The shape restrictions to be studied include monotone, convex, log-concave, and log-convex functions. He aims to develop new basic empirical process tools and methods and to apply the new tools to problems involving shape-restricted inference, semiparametric models, bootstrap inference methods, and problems in high-dimensional statistics. The investigator intends also to introduce and evaluate new methods of estimation in semiparametric models based on complex sampling designs with missing data by design, and for estimation in semiparametric models defined by shape constraints.