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LOG-CONCAVITY OF COMPOUND DISTRIBUTIONS WITH APPLICATIONS IN OPERATIONAL AND ACTUARIAL MODELS

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  23 August 2019

F. G. Badía ,
C. Sangüesa Open the ORCID record for C. Sangüesa [Opens in a new window]  and
A. Federgruen
F. G. Badía
Affiliation:
Department of Statistical Methods and IUMA, University of Zaragoza, Zaragoza50018, Spain E-mail: gbadia@unizar.es; csangues@unizar.es
C. Sangüesa
Affiliation:
Department of Statistical Methods and IUMA, University of Zaragoza, Zaragoza50018, Spain E-mail: gbadia@unizar.es; csangues@unizar.es
A. Federgruen
Affiliation:
Graduate School of Business, Columbia University, New York10027, United States E-mail: af7@gsb.columbia.edu
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Abstract

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.

We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

Keywords

actuarial modelcompound distributioninventory modellog-concavityreplacement model

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 2 , April 2021 , pp. 210 - 235
Copyright
Copyright © Cambridge University Press 2019

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