Univariate likelihood projections and characterizations of the multivariate normal distribution

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 displayed 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, classical multivariate normality tests, such as Shapiro–Wilk's, Henze–Zirkler's and Mardia's tests, may break down completely.