Robust Statistical Inference for High-Dimension Low Sample Size Models with Application to Genomics

Abstract

In high-dimension (K) low sample size (n) environments, often, nonlinear, inequality, order, or general shape constraints crop up in rather complex ways. As a result, likelihood principle based optimal statistical inference procedures either may not be in a closed, manageable form or even may not exist. While some of these complex statistical inference problems can be treated in suitable asymptotic setups, the curse of dimensionality (i.e., K > > n, with n possibly small) calls for a different asymptotics route (K very large) having different perspectives. Roy's (1953) union-intersection principle, with genesis in the likelihood principle, provides some alternative approaches which are generally more amenable for the K > > n environments. This scenario is appraised with two important applications in genomics (microarray data and SNP data models) where a large number (K) of genes with plausible dependence as well as heterogeneity amidst a small sample size environment creates impasses for standard robust statistical inference. These statistical perspectives are appraised in some nonstandard ways.