Increasing hazard rate of mixtures for natural exponential families

Shaul K. Bar-Lev, Gérard Letac

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)373-390
Number of pages18
JournalAdvances in Applied Probability
Issue number2
StatePublished - Jun 2012


  • Cubic variance function
  • Hyperbolic cosine NEF
  • Kummer type-2 NEF
  • Mixture
  • Natural exponential family (NEF)
  • Quadratic variance function
  • Ressel NEF
  • Variance function

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics


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