Asymptotic Properties of the Hazard Function and the Mean Residual Life Function

The memoryless or non-aging property of systems is of special relevance in reliability theory, which implies that the hazard function is constant in time, and the corresponding mean residual life function takes a reciprocal value. The only known continuous distribution with that property is the exponential distribution. However, many other distributions exist whose asymptotic behavior of underlying hazard functions approaches a constant, while the mean residual life function approaches a reciprocally constant value. Here we provide an analysis which enables us to study a class of distributions that asymptotically approach the memoryless property, and which include gamma, Erlangian, exponential resilience, exponential geometric, hyper exponential, logistic exponential and the inverse Gaussian distribution.

SHARP BOUNDS FOR SURVIVAL PROBABILITY WHEN AGEING IS NOT MONOTONE

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

MEAN RESIDUAL LIFE FUNCTION FOR ADDITIVE AND MULTIPLICATIVE HAZARD RATE MODELS

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

Reliability Analysis of the Proportional Mean Residual Life Order

The concept of mean residual life plays an important role in reliability and life testing. In this paper, we introduce and study a new stochastic order called proportional mean residual life order. Several characterizations and preservation properties of the new order under some reliability operations are discussed. As a consequence, a new class of life distributions is introduced on the basis of the anti-star-shaped property of the mean residual life function. We study some reliability properties and some characterizations of this class and provide some examples of interest in reliability.