Analytical results for IGFR and DRPFR distributions, with actuarial applications

Eva María Ortega and Xiaohu Li

Communications in Statistics - Theory and Methods, 2019, vol. 48, issue 2, 178-202

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.

Date: 2019
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DOI: 10.1080/03610926.2015.1082591

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