Asymptotic power of Rao's score test for independence in high dimensions

Abstract

Let $\mathbf{R}$ be the Pearson correlation matrix of $m$ normal random variables. The Rao’s score test for the independence hypothesis $H_{0}:\mathbf{R}=\mathbf{I}_{m}$, where $\mathbf{I}_{m}$ is the identity matrix of dimension $m$, was first considered by Schott (Biometrika 92 (2005) 951–956) in the high dimensional setting. In this paper, we study the exact power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Rao’s score test is minimax rate-optimal for detecting the depen..