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A note on strong unimodality and dispersivity

Published online by Cambridge University Press:  14 July 2016

W. Droste  and
W. Wefelmeyer
W. Droste*
Affiliation:
University of Cologne
W. Wefelmeyer*
Affiliation:
University of Cologne
*
∗Postal address: Institute of Mathematics, University of Cologne, Weyertal 86, D-5000 Cologne 41, West Germany.
∗Postal address: Institute of Mathematics, University of Cologne, Weyertal 86, D-5000 Cologne 41, West Germany.
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Abstract

A distribution R is ‘more dispersed' than a distribution P if any two quantiles of R are more widely separated than the corresponding quantiles of P. Lewis and Thompson (1981) prove that a non-degenerate distribution P is strongly unimodal if and only if it is ‘dispersive', i.e. if P∗ Q1 is more dispersed than P∗ Q2 whenever Q1 is more dispersed than Q2. For P admitting a positive Lebesgue density, we prove that P is strongly unimodal if and only if P ∗ Q is more dispersed than P for every distribution Q. Hence the latter property also characterizes the dispersivity of P.

Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 235 - 239
Copyright
Copyright © Applied Probability Trust 1985 

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