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Increasing Hazard Rate of Mixtures for Natural Exponential Families

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  04 January 2016

Shaul K. Bar-Lev  and
Gérard Letac
Shaul K. Bar-Lev*
Affiliation:
University of Haifa
Gérard Letac*
Affiliation:
Université Paul Sabatier
*
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel. Email address: [email protected]
∗∗ Postal address: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France. Email address: [email protected]
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Abstract

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Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

Keywords

Natural exponential family (NEF)mixturevariance functionquadratic variance functioncubic variance functionhyperbolic cosine NEFRessel NEFKummer type-2 NEF

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 373 - 390
Copyright
© Applied Probability Trust 

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