Unified Loss Function and Estimating Function Based Learning

Current applications in genomics and epidemiology concern high dimensional (and, possibly, time-dependent) data structures, and the questions of interest correspond typically with high dimensional parameters of interest. In such problems it is typically not possible to a priory pose a model allowing estimation at a parametric rate, and thereby requiring estimators of non-pathwise differentiable parameters. We will present a general loss based estimation procedure, which is grounded by theory (e.g., minimax adaptive), and generalizes existing estimation problems. An application of this methodology yields data adaptive algorithms for conditional mean estimation, conditional hazard/density estimation based on censored and uncensored data. In addition, we present a general estimating function based estimation procedure for pathwise and non-pathwise differentiable parameters. Both methodologies involve loss based and estimating function based cross-validation as a tool to select among candidate estimators of the parameter of interest. We illustrate the methodology with some applications in genomics and epidemiology.