Robust Statistical Inference for High-Dimensional Data 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 complex ways, and as a result, likelihood based optimal statistical inference procedures may not exist, at least, may not be in manageable form. While some of these inference problems can be treated in asymptotic setups, the curse of dimensionality (i.e., K >> n with often n small) calls for a different type of asymptotics (in K) with different perspectives. Roy’s union-intersection principle provides some alternative approaches, generally more amenable for K >> n environments. This scenario is appraised with two important statistical problems in genomic studies: a large number of (possibly dependent) genes with heterogeneity amidst a smaller sample create impasses for standard robust inference. These perspectives are examined here in a nonstandard statistical analysis.