Distribution of the Sample Correlation Matrix and Applications

Thu Pham-Gia,Vartan Choulakian

doi:10.4236/ojs.2014.45033

Abstract

For the case where the multivariate normal population does not have null correlations, we give the exact expression of the distribution of the sample matrix of correlations R, with the sample variances acting as parameters. Also, the distribution of its determinant is established in terms of Meijer G-functions in the null-correlation case. Several numerical examples are given, and applications to the concept of system de- pendence in Reliability Theory are presented.

Highlights

The correlation matrix plays an important role in multivariate analysis since by itself it captures the pairwise degrees of relationship between different components of a random vector
Component Analysis and Factor Analysis, where in general, it gives results different from those obtained with the covariance matrix
The distribution of the sample correlation matrix R is relatively easy to compute, and its determinant has a distribution that can be expressed as a Meijer G-function distribution

Summary

Introduction

The correlation matrix plays an important role in multivariate analysis since by itself it captures the pairwise degrees of relationship between different components of a random vector. Component Analysis and Factor Analysis, where in general, it gives results different from those obtained with the covariance matrix. The distribution of the sample correlation matrix R is relatively easy to compute, and its determinant has a distribution that can be expressed as a Meijer G-function distribution. When Λ ≠ I no expression for the density of R is presently available in the literature, and the distribution of its determinant is still unknown, in spite of efforts made by several researchers. We will provide here the closed form expression of the distribution of R, with the sample variances as parameters, complementing a result presented by Fisher in [2]

Objectives

Results

Conclusion