Advances in Applied Probability

Increasing hazard rate of mixtures for natural exponential families

Shaul K. Bar-Lev and Gérard Letac

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 2 (2012), 373-390.

Dates
First available in Project Euclid: 16 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1339878716

Digital Object Identifier
doi:10.1239/aap/1339878716

Mathematical Reviews number (MathSciNet)
MR2977400

Zentralblatt MATH identifier
1246.60029

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60E05: Distributions: general theory

Keywords
Natural exponential family (NEF) mixture variance function quadratic variance function cubic variance function hyperbolic cosine NEF Ressel NEF Kummer type-2 NEF

Citation

Bar-Lev, Shaul K.; Letac, Gérard. Increasing hazard rate of mixtures for natural exponential families. Adv. in Appl. Probab. 44 (2012), no. 2, 373--390. doi:10.1239/aap/1339878716. https://projecteuclid.org/euclid.aap/1339878716


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