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Dispersive distributions, and the connection between dispersivity and strong unimodality

Published online by Cambridge University Press:  14 July 2016

Toby Lewis  and
J. W. Thompson
Toby Lewis
Affiliation:
University of Hull
J. W. Thompson*
Affiliation:
University of Hull
*
∗∗Postal address: Department of Mathematical Statistics, The University, Hull, U.K.
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Abstract

Two continuous distributions, G, H so related that any two quantiles of H are more widely separated than the corresponding quantiles of G may be said to be ‘ordered in dispersion'; Saunders and Moran have given examples. It is shown here that distributions F (called ‘dispersive' distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, F ∗H. The class of dispersive distributions is determined, and shown to coincide with the class of strongly unimodal distributions.

Keywords

QUANTILEORDERING IN DISPERSIONDISPERSIVITYSTRONG UNIMODALITYCONVOLUTIONINFINITE DIVISIBILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 76 - 90
Copyright
Copyright © Applied Probability Trust 

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Footnotes

∗

Present address: Faculty of Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, U.K.

Research carried out while this author held a visiting appointment in the CSIRO Division of Mathematics and Statistics, Canberra, Australia.

References

Ibragimov, I. A. (1956) On the composition of unimodal distributions. Theory Prob. Appl. 1, 255–260.CrossRefGoogle Scholar
Lukacs, E. (1970) Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
Saunders, I. W. (1978) Locating bright spots in a point process. Adv. Appl. Prob. 10, 587–612.CrossRefGoogle Scholar
Saunders, I. W. and Moran, P. A. P. (1978) On the quantiles of the gamma and F distributions. J. Appl. Prob. 15, 426–432.Google Scholar