Some bounds related to Harris family of distributions

Abstract

As a lifetime distribution, Harris family of distributions are applied to the lifetime of a series system with random number of components. In this paper, properties of various ageing classes of mixtures of Harris family of distributions, where the tilt parameter of a Harris distribution is taken as a random variable, are studied. We obtain an upper bound for maximum error in evaluating its reliability function. Two bounds are also presented for survival function and expectation of the mixed Harris family. We also provide some interesting bounds for its residual survival function. Our results generalize several previous findings in this connection. Some illustrative examples are also provided.

Keywords:

• Ageing
• Marshall–Olkin distribution
• harmonically new better than used
• mean residual lifetime
• reliability function

Acknowledgement

The authors would like to thank two anonymous reviewers whose valuable comments led to an improved version of the manuscript.