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NWU property of a class of random sums

Part of: Special processes Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Jun Cai  and
Vladimir Kalashnikov
Jun Cai*
Affiliation:
University of Waterloo
Vladimir Kalashnikov*
Affiliation:
Institute for Information Transmission Problems
*
∗Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
∗∗Postal address: Institute for Information Transmission Problems, Bol'shoy Karetny 19, 101447 Moscow, Russia
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Abstract

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

Keywords

NWU distributiondiscrete NWU distributiongeometric sumrandom sumrenewal processmixed Poisson distribution

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 283 - 289
Copyright
Copyright © 2000 by The Applied Probability Trust 

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