Efficient Estimators for Expectations in Nonlinear Parametric Regression Models with Responses Missing at Random & Data Integration in High Dimension with Multiple Quantiles

Abstract

This dissertation contains the two research projects in my Ph.D. study. The first project considers nonlinear regression models that are solely defined by a parametric model for the regression function. The responses are assumed to be missing at random, with the missingness depending on multiple covariates. We propose estimators for expectations of a known function of response and covariates. Our estimator is a nonparametric estimator corrected for the regression function. We show that it is asymptotically efficient in the Hajek and Le Cam sense. ´ Simulations and an example using real data confirm the optimality of our approach. The second project deals with aggregating and analyzing high dimensional data, which come from multiple experiments and thus have different responses, but share the same predictors. The measurements of the predictors may be different across experiments. In each experiment multiple conditional quantiles are considered simultaneously, assuming a linear relationship between the response and predictors. To select the predictors that affect any of the responses at any of the quantile levels, we propose a penalized estimation process and an information criterion and study the asymptotic properties. Simulations and a real data application demonstrate the advantage of combining information from multiple experiments and quantile levels.