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July, 1978 A Monotone Unimodal Distribution which is Not Central Convex Unimodal
Daniel R. Wells
Ann. Statist. 6(4): 926-931 (July, 1978). DOI: 10.1214/aos/1176344268


For symmetric univariate distributions the usual definition of unimodality due to Khintchine has several equivalent formulations. When these concepts are generalized to higher dimensions in an attempt to define multivariate unimodality questions concerning their equivalence naturally arise. Of particular interest in this area is the relationship between two concepts first studied by Sherman and more recently by Dharmadhikari and Jogdeo. They asked if requiring that a distribution belong to the closed convex hull of all uniform distributions on symmetric convex bodies was the same as requiring that the probability it assigns to a symmetric convex set decrease as the set is translated away from the origin in a fixed direction. Sherman conjectured that the two concepts were the same while Dharmadhikari and Jogdeo felt that this was not so and they suggested a possible counterexample to Sherman's conjecture. In this paper it is shown that their example is indeed a counterexample.


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Daniel R. Wells. "A Monotone Unimodal Distribution which is Not Central Convex Unimodal." Ann. Statist. 6 (4) 926 - 931, July, 1978.


Published: July, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0378.62013
MathSciNet: MR494406
Digital Object Identifier: 10.1214/aos/1176344268

Primary: 52A40
Secondary: 62E10

Keywords: central convex unimodality , convexity , monotone unimodality

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • July, 1978
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