Univariate Likelihood Projections and Characterizations of the Multivariate Normal Distribution

TL;DR

This paper explores how univariate likelihood projections can uniquely characterize multivariate normal distributions and proposes new testing methods that outperform classical tests in certain non-normal scenarios.

Contribution

It introduces a novel approach to characterize multivariate normality using univariate likelihood projections and develops robust testing techniques based on this principle.

Findings

Univariate likelihood projections can uniquely identify multivariate normal distributions.

The proposed tests show higher power than traditional methods in non-MN cases.

Classical tests may fail when components are normal but the joint distribution is not.

Abstract

The problem of characterizing a multivariate distribution of a random vector using examination of univariate combinations of vector components is an essential issue of multivariate analysis. The likelihood principle plays a prominent role in developing powerful statistical inference tools. In this context, we raise the question: can the univariate likelihood function based on a random vector be used to provide the uniqueness in reconstructing the vector distribution? In multivariate normal (MN) frameworks, this question links to a reverse of Cochran's theorem that concerns the distribution of quadratic forms in normal variables. We characterize the MN distribution through the univariate likelihood type projections. The proposed principle is employed to illustrate simple techniques for assessing multivariate normality via well-known tests that use univariate observations. The presented testing strategy can exhibit high and stable power characteristics in comparison to the well-known procedures in various scenarios when observed vectors are non-MN distributed, whereas their components are normally distributed random variables. In such cases, the classical multivariate normality tests may break down completely. KEY WORDS: Characterization, Goodness of fit, Infinity divisible, Likelihood, Multivariate normal distribution, Projection, Quadratic form, Test for multivariate normality.