Title: Contiguity under high-dimensional Gaussianity with applications to covariance testing
Authors: Qiyang Han - Rutgers University (United States)
Tiefeng Jiang - University of Minnesota (United States)
Yandi Shen - University of Chicago (United States) [presenting]
Abstract: Le Cams third/contiguity lemma is a fundamental probabilistic tool in mathematical statistics for several major developments in classical statistical estimation and testing theories. Despite widespread applications to low-dimensional statistical problems, the stringent requirement of Le Cams third/contiguity lemma on the asymptotic distributional expansions of the log-likelihood makes it challenging to use in many modern high-dimensional statistical problems. A non-asymptotic analogue of Le Cams third/contiguity lemma is established under high dimensional normal populations, which requires only mean and variance bounds of the statistic under study, but without an exact distributional expansion of the likelihood ratio. As a demonstration of the power of the new contiguity result, we obtain asymptotically exact power formulae for a number of widely used high-dimensional covariance tests, including the likelihood ratio tests and trace tests, that hold uniformly over all possible alternatives under mild growth conditions on the dimension-to-sample ratio. These new results go far beyond the scope of previous available case-specific techniques, and exhibit new phenomena regarding the behavior of these important class of covariance tests.