Analytical results for IGFR and DRPFR distributions, with actuarial applications

Abstract

An increasing generalized failure rate of a lifetime X defines an ageing concept, denoted by IGFR. Another notion, denoted by DRPFR, is defined by the decreasingness of the reversed proportional failure rate. In this article, we provide characterizations for both IGFR and DRPFR absolutely continuous lifetimes, based on monotonicity of quotients of probabilistic functionals and a result by Nanda and Shaked (2001). We derive the necessary conditions for the IGFR notion, based on stochastic orderings of truncated distributions, and we prove that the product of DRPFR lifetimes is also DRPFR; that the IGFR property is preserved by composition with certain risk aversion utility functions; and that the order statistics and the records (and the subsequent order statistic (record)) are IGFR under suitable assumptions, with similar results for DRPFR lifetimes. Also, we provide sufficient conditions for the hazard rate ordering of products and random products of IGFR lifetimes, and similar results for the reversed hazard rate order and DRPFR lifetimes, with a complementary result for the mean residual life order of random products of two families of IGFR lifetimes, we derive the upper and lower bounds for the cumulative distribution function of the product of IGFR lifetimes, and we provide the lower bounds for the risk function of an IGFR lifetime based on the distribution moments, and these bounds are extended for the product of IGFR lifetimes. We discuss extensively the applications of the results in insurance portfolios.

Keywords:

• Elasticity function;
• Generalized failure rate;
• Hazard rate ordering;
• Income distribution;
• Option pricing;
• Order statistics;
• Records;
• Reversed hazard rate ordering.

MATHEMATICS SUBJECT CLASSIFICATION:

• 60E15;
• 62P05

Acknowledgments

Thanks are due to an anonymous referee for the ideas to improve the state-of-the-art of the application of the notions EM. Ortega is sincerely grateful to Prof. Franco Pellerey and Prof. German Badia, and also to her friends Manuela, Mary Carmen, and Marisa for their encouragement during the research. The article was presented at the XXXII National Conference on Statistics and Operations Research, SEIO2010, in Coruña, Spain. Thanks are due also to my public funds with reference MTM2009.13433 from Plan Nacional Investigacion Desarrollo e Innovacion Competitive Call 2009 Type Individual, to finance my One-Person Research Project that ended in 2011. This article was submitted firstly in the year 2010.

Additional information

Funding

The results in this article are financed with the funds with reference in the Acknowledgements.