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Some characterizations of exponential or geometric distributions in a non-stationary record value model

Part of: Singular integral equations Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

Hans-Jürgen Witte
Hans-Jürgen Witte*
Affiliation:
University of Oldenburg
*
∗ Postal address: Carl von Ossietzky Universität Oldenburg, Fachbereich 6 Mathematik, Postfach 2503, DW-2900 Oldenburg, Germany.
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Abstract

Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.

Keywords

EXPONENTIAL TAIL DISTRIBUTIONSGEOMETRIC (TAIL) DISTRIBUTIONSRECORD VALUES IN A NON-STATIONARY MODELORDER STATISTICSINTEGRATED CAUCHY FUNCTIONAL EQUATIONCHARACTERIZATION PROBLEMS

MSC classification

Primary: 60E05: Distributions
Secondary: 62E10: Characterization and structure theory 45E10: Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 373 - 381
Copyright
Copyright © Applied Probability Trust 1993 

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