High-Dimensional Edgeworth Expansion of LR Statistic for Testing Circular Symmetric Covariance Structure and Its Error Bound

Abstract

This article is concerned with the null distribution of a likelihood ratio (LR) statistic for testing circular symmetric covariance structure. We give an asymptotic expansion of the null distribution under the assumption of normality as the number of variables p and the sample size N go to infinity together with the ratio p/N is converging on a finite non zero constant limit c ∈ (0, 1). Numerical simulations reveal that our approximation is more accurate than the classical χ2-type approximation as p increase in value. Furthermore, we derive a computable error bound for its asymptotic expansions.