The investigator carries out research on problems involving shape restricted inference, empirical process methods and tools, and new theory for semiparametric and nonparametric models. In particular, the investigator and a University of Washington graduate student conduct research on new confidence set procedures in the context of convex function estimation and new semiparametric models involving shape restrictions. The investigator and a University of Washington graduate biostatistics graduate student conduct research on improved estimation methods for semiparametric models with two-phase designs with missing data (by design). This research also involves development of new asymptotic theory for a variety of complex sampling methods and multi-phase designs. Some of the research on empirical process methods and tools is carried out jointly with colleagues in the Netherlands. Some of the research on inference under shape constrained estimation is carried out jointly with colleagues in France, Switzerland, and Canada. These investigations involve nonstandard asymptotics for maximum likelihood estimators, likelihood ratio statistics, and new nonstandard limit distributions. Part of the proposed research involves better understanding of bootstrap and other resampling procedures in high-dimensional settings. The research also involves 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 high-dimensional data. One key goal involves improved theory for empirical likelihood and generalized empirical likelihood estimation methods. Another goal is to understand the effect of heavier tailed distributions in high-dimensional statistical problems.
Applications include regression models with high-dimensional covariates, models for survival data with missing covariate data, and non- and semi-parametric maximum likelihood estimators used in HIV-AIDS research. 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 graduate level courses for the Departments of Statistics and Biostatistics at the University of Washington.