On testing mean proportionality of multivariate normal variables

TL;DR

This paper proves that the likelihood ratio chi-squared test for mean proportionality in multivariate normal variables is valid even in finite samples, using eigenvalue and polynomial representations.

Contribution

It demonstrates the non-asymptotic validity of the likelihood ratio test for mean proportionality in multivariate normal distributions.

Findings

Likelihood ratio test is valid non-asymptotically

Test statistic can be expressed as the minimum eigenvalue of a Wishart matrix

Distribution representation involves Legendre polynomials

Abstract

This short note considers the problem of testing the null hypothesis that the mean values of two multivariate normal variables are proportional. We show that the usual likelihood ratio $\chi^2$-test is valid non-asymptotically. Our proof relies on expressing the test statistic as the minimum eigenvalue of a Wishart variable and using a representation of its distribution using Legendre polynomials.