Advances in Applied Probability

Increasing hazard rate of mixtures for natural exponential families

Shaul K. Bar-Lev and Gérard Letac

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 16 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • Abad, J. and Sesma, J. (1995). Computation of the regular confluent hypergeometric function. Mathematica J. 5, 74–76.
  • Barlow, R. E. and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley, New York.
  • Ben Salah, N. and Masmoudi, A. (2011). The real powers of the convolution of a gamma distribution and a Bernoulli distribution. J. Theoret. Prob. 24, 450–453.
  • Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behavior of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740.
  • Block, H. W., Li, Y. and Savits, T. H. (2005). Mixtures of normal distributions: modality and failure rate. Statist. Prob. Lett. 74, 253–264.
  • Fitzgerald, D. L. (2002). Tricomi and Kummer functions in occurrence, waiting times and exceedance statistics. Stoch. Environ. Res. Risk Assess. 16, 207–214.
  • Fosam, E. B. and Shanbhag, D. N. (1997). An extended Laha-Lukacs characterization result based on a regression property. J. Statist. Planning Infer. 63, 173–186.
  • Glaser, R. E. (1980). Bathtub and related failure rate characterization. J. Amer. Statist. Assoc. 75, 667–672.
  • Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products. Academic Press, New York.
  • Karlin, S. (1968). Total Positivity, Vol. 1. Stanford University Press.
  • Karlin, S. and Proschan, F. (1960). Pólya type distributions of convolutions. Ann. Math. Statist. 31, 721–736.
  • Kokonendji, C. C. (2001). First passage times on zero and one for natural exponential families. Statist. Prob. Lett. 51, 293–298.
  • Letac, G. and Mora, M. (1990). Natural real exponential families with cubic variance functions. Ann. Statist. 18, 1–37.
  • Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65–80.
  • Navarro, J. and Hernandez, P. J. (2004). How to obtain bathtub-shaped failure rate models from normal mixtures. Prob. Eng. Inf. Sci. 18, 511–531.
  • Navarro, J., Guillamón, A. and Ruiz, M. C. (2009). Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Appl. Stoch. Models Business Industry 25, 323–337.
  • Ng, K. W. and Kotz, S. (1995). Kummer-Gamma and Kummer-Beta univariate and multivariate distributions. Res. Rep., Department of Statistics, The University of Hong Kong.
  • Pakes, A. G. (1996). A hitting time for Lévy processes, with applications to dams and branching processes. Ann. Fac. Sci. Toulouse Math. 5, 521–544.
  • Prabhu, N. U. (1965). Queues and Inventories. A Study of Their Basic Stochastic Processes. John Wiley, New York.
  • Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • Sibuya, M. (2006). Applications of hyperbolic secant distributions. Japanese J. Appl. Statist. 35, 17–47.
  • Shanbhag, D. N. (1979). Diagonality of the Bhattacharyya matrix as a characterization. Theory Prob. Appl. 24, 430–433.
  • Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.
  • Smyth, G. K. (1994). A note on modelling cross correlations: hyperbolic secant regression. Biometrika 81, 396–402.
  • Zolotarev, V. M. (1967). On the divisibility of stable laws. Theory Prob. Appl. 12, 506–508. \endharvreferences