A New Family of Multivariate Distributions with Applications to Monte Carlo Studies

Abstract

Abstract We consider a new class of multivariate probability distributions having representation X = RLY (p), where R is distributed as √χ 2 (p), L is the Choleski factorization of the scaling matrix Σ, and Y (p) represents an arbitrary distribution on the p-dimensional unit hypersphere. If Y (p) is uniform, then X has a multivariate normal distribution with mean 0 and covariance matrix Σ. The use of classical spherical distributions or other nonuniform distributions for Y (p) leads to interesting, controllable departures from normality that are particularly relevant to robustness studies. Their use is illustrated in a Monte Carlo investigation of the robustness of Hotelling's T 2. Variate generation routines for the cardioid, triangular, offset normal, wrapped normal, wrapped Cauchy, von Mises, power sine, Fisher, and Bingham distributions are developed.