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May, 1976 Multivariate Unimodality
S. W. Dharmadhikari, Kumar Jogdeo
Ann. Statist. 4(3): 607-613 (May, 1976). DOI: 10.1214/aos/1176343466


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.


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S. W. Dharmadhikari. Kumar Jogdeo. "Multivariate Unimodality." Ann. Statist. 4 (3) 607 - 613, May, 1976.


Published: May, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0338.62006
MathSciNet: MR415719
Digital Object Identifier: 10.1214/aos/1176343466

Primary: 52A40
Secondary: 62E10

Keywords: convexity , convolutions and weak limits of unimodal distributions , definitions of multivariate unimodality

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • May, 1976
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