# Test of Covariance and Correlation Matrices

## TL;DR

This paper introduces a flexible, distribution-free framework for testing covariance and correlation matrices using a generalized cosine measure, applicable to various one-sample and two-sample scenarios with real data and simulations.

## Contribution

It develops a novel, distribution-agnostic testing framework for covariance and correlation matrices, including permutation algorithms for common tests, accommodating diverse data distributions and dimensions.

## Key findings

- Performs well in controlling type I error across tests.
- Shows high power in detecting covariance and correlation differences.
- Applicable to data with different marginal distributions.

## Abstract

Based on a generalized cosine measure between two symmetric matrices, we propose a general framework for one-sample and two-sample tests of covariance and correlation matrices. We also develop a set of associated permutation algorithms for some common one-sample tests, such as the tests of sphericity, identity and compound symmetry, and the $K$-sample tests of multivariate equality of covariance or correlation matrices. The proposed method is very flexible in the sense that it does not assume any underlying distributions and data generation models. Moreover, it allows data to have different marginal distributions in both the one-sample identity and $K$-sample tests. Through real datasets and extensive simulations, we demonstrate that the proposed method performs well in terms of empirical type I error and power in a variety of hypothesis testing situations in which data of different sizes and dimensions are generated using different distributions and generation models.