Increasing hazard rate of mixtures for natural exponential families

TL;DR

This paper establishes a sufficient condition for mixtures within the same natural exponential family to exhibit increasing hazard rates, with applications to specific NEFs like hyperbolic cosine, Ressel, and Kummer distributions.

Contribution

It provides a new sufficient condition for increasing hazard rates in mixtures of NEF distributions, with detailed applications to complex NEFs.

Findings

Condition verified for hyperbolic cosine NEF

Condition verified for Ressel NEF

Condition verified for Kummer distributions of type 2

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 a 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 NEF's having either quadratic, or cubic variance functions (VF) and others as well. A particular attention is devoted to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF and to the Kummer distributions of type 2 NEF. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEF's. Various lemmas and propositions are needed then to verify this condition for these NEF's.