A new class of multivariate goodness of fit tests for multivariate normal mixtures

Abstract

The goodness of fit (GOF) test plays an important role to determine whether a data set comes from an assumed distribution (with possibly unknown parameters). Many approaches have been proposed to perform the goodness of fit test, such as Chi-square, Kolmogorov-Smirnov, Cramér-Von Mises, and Anderson Darling test statistics, but most of those are proposed for univariate data. In this paper, we propose a new class of goodness of fit tests for assessing multivariate normal mixtures (with multivariate normality as a special case) based on dimension reduction. The new method allows us to directly employ existing univariate GOF tests to perform multivariate GOF tests after linearly projecting multivariate data to univariate data. The new GOF statistic is defined as the maximum of some one-dimensional GOF test statistics. We propose a bootstrap procedure to approximate the distribution of the new GOF statistic. Our approach is simple, general and can be easily extended to perform GOF tests for many other multivariate distributions. The effectiveness of our proposed new GOF tests for multivariate normal mixture models is demonstrated via simulation studies and a real data application.