Shape Restrictions, Semiparametric Models, and Empirical Processes

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Jon A. Wellner will carry out research on empirical process methods and computational strategies for a variety of semiparametric and nonparametric models for shape restrictions, censored data, and missing data problems.In particular, Wellner and University of Washington graduate student Marios Pavlides will conduct research on several models involving multivariate monotone density functions and multivariate current status observation schemes, a class of models currently of interest in connection with HIV-AIDS studies. Wellner and University of Washington graduate student Arseni Seregin will conduct research on nonparametric estimation of log-concave densities and generalizations thereof in one and higher dimensions. Some of the research on empirical processes will be carried out jointly with Aad van der Vaart (Free University, Amsterdam). Some of the research on inference under shape constraints will be carried out jointly with Fadoua Balabdaoui, Marloes Maathuis, and Shuguang Song (former Ph.D. students). The work on new computational algorithms will be carried out jointly with Hanna Jankowski, a recent post-doctoral student at the University of Washington and now a faculty member at York University, Canada. These investigations will involve nonstandard asymptotics for maximum likelihood estimators, likelihood ratio statistics, and distribution theory for new nonstandard limit distributions.The research will involve development of basic empirical process tools and methods, and applications of these new tools and methods to statistical problems concerning semiparametric models, shape restricted models, and inverse problems. Applications include regression models for panel count data, bivariate interval censored data of several kinds, regression models for survival data with missing covariate data, studies of non- and semi-parametric maximum likelihood estimators used in AIDS research, new confidence bands for shape restricted inference, and both basic empirical process theory and new semiparametric estimation procedures for two-phase data dependent designs. The advances in inference methods for competing risks models are used to study causative effects of viral loading in HIV vaccine trials. The developments in bivariate censored data are applied to studies of mother-to-child transmission of HIV-AIDS. The work on two-phase data dependent designs has application to new designs with increased efficiency for clinical trials and case-cohort sampling in epidemiology. The tools of empirical process theory allow investigations of many problems of current interest in other areas of statistics involving high-dimensional data and parameter spaces. The research benefits education and human development by the training of graduate students and the inclusion of the resulting new statistical methods in statistics courses at the University of Washington and elsewhere.