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

## TL;DR

This paper analyzes the asymptotic power of Rao's score test for independence in high-dimensional normal data, showing it is rate-optimal for detecting dependencies as both sample size and dimension grow.

## Contribution

It derives the asymptotic minimax power function of Rao's score test in high dimensions, establishing its rate-optimality for dependency detection.

## Key findings

- Rao's score test is rate-optimal for dependency signals of order sqrt(m/n)
- The test's power function is characterized asymptotically in high dimensions
- Both dimension and sample size tend to infinity with bounded ratio

## Abstract

Let ${\bf R}$ be the Pearson correlation matrix of $m$ normal random variables. The Rao's score test for the independence hypothesis $H_0 : {\bf R} = {\bf I}_m$, where ${\bf I}_m$ is the identity matrix of dimension $m$, was first considered by Schott (2005) in the high dimensional setting. In this paper, we study the asymptotic minimax 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 rate-optimal for detecting the dependency signal $\|{\bf R} - {\bf I}_m\|_F$ of order $\sqrt{m/n}$, where $\|\cdot\|_F$ is the matrix Frobenius norm.