Empirical likelihood method for complete independence test on high dimensional data

TL;DR

This paper introduces a novel empirical likelihood method for testing mutual independence in high-dimensional data, providing theoretical distribution results and demonstrating improved power through simulations.

Contribution

It develops a one-sided empirical likelihood approach for independence testing in high dimensions and introduces a rescaled version to enhance test power.

Findings

The limiting distribution of the test statistic is derived as $Z^2I(Z>0)$.

The rescaled empirical likelihood test shows improved power in simulations.

The method is effective for large $n$ and $p$ in multivariate normal data.

Abstract

Given a random sample of size $n$ from a $p$ dimensional random vector, where both $n$ and $p$ are large, we are interested in testing whether the $p$ components of the random vector are mutually independent. This is the so-called complete independence test. In the multivariate normal case, it is equivalent to testing whether the correlation matrix is an identity matrix. In this paper, we propose a one-sided empirical likelihood method for the complete independence test for multivariate normal data based on squared sample correlation coefficients. The limiting distribution for our one-sided empirical likelihood test statistic is proved to be $Z^2I(Z>0)$ when both $n$ and $p$ tend to infinity, where $Z$ is a standard normal random variable. In order to improve the power of the empirical likelihood test statistic, we also introduce a rescaled empirical likelihood test statistic. We carry out an extensive simulation study to compare the performance of the rescaled empirical likelihood method and two other statistics which are related to the sum of squared sample correlation coefficients.