Multivariate Unimodality

Abstract

For univariate distributions there is a generally accepted definition of unimodality due to Khintchine which requires the existence of a number $a$, called vertex, such that the distribution function is convex on $(-\infty, a)$ and concave on $(a, \infty)$. For multivariate distributions, however, unimodality can be defined in several different ways. Anderson (1955) and Olshen and Savage (1970) have given such definitions. This paper first examines the definition which calls a random vector $(X_1, \cdots, X_n)$ unimodal if all linear combinations $\sum a_iX_i$ are univariate unimodal. We show that this definition is somewhat unnatural because the density of such a "unimodal" distribution may not become maximum at the vertex of unimodality. The paper also examines two other definitions based on the results of Sherman (1955). One of these looks at the closed convex hull of the set of all uniform distributions on symmetric convex bodies and the other requires that the probability carried by a symmetric convex set decreases as the set is moved away from the origin in a fixed direction. The equivalence of these definitions was conjectured by Sherman and this paper gives some results having a bearing on this conjecture.