Some results on the proportional reversed hazards model

doi:10.1016/S0167-7152(00)00127-9

Statistics & Probability Letters

Volume 50, Issue 4, 1 December 2000, Pages 313-321

https://doi.org/10.1016/S0167-7152(00)00127-9 Get rights and content

Abstract

The proportional reversed hazards model consists in describing random failure times by a family ${[F(x)]^{θ}, θ>0}$ of distribution functions, where F(x) is a baseline distribution function. We show various results on this model related to some topics in reliability theory, including ageing notions of random lifetimes, comparisons based on stochastic orders, and relative ageing of distributions.

Introduction

The well-known proportional hazards model is expressed by the following relation between survival functions of random lifetimes: $F ̄_{X̂} (x)=[F ̄_{X} (x)]^{θ}, x∈ R$ , θ>0. Recently, Gupta et al. (1998) have proposed a dual model, called proportional reversed hazards model. The latter is expressed by a similar relation between distribution functions: $F_{X^{∗}} (x)=[F_{X} (x)]^{θ}$ , with $x∈ R$ and θ>0, so that the reversed hazard rate functions of lifetimes X and $X^{∗}$ are proportional, i.e. $τ_{X^{∗}} (x)=θτ_{X} (x)$ .

In this paper we present some results on the proportional reversed hazards model which are of interest in reliability theory.

In Section 2 we describe the basic aspects of the model, including some examples.

In Section 3 we prove the following preservation results of ageing properties:

Some stochastic comparisons are given in Section 4. Indeed, we show suitable sufficient conditions such that $θ X$ and $X^{∗}$ are ordered according to ⩽_st-order. We also face the problem of preservation of some stochastic orders under the transformation $X→X^{∗}$ .

Finally, a connection between the proportional reversed hazards model and some notions of relative ageing of distributions is discussed in Section 5. In particular, denoting by $T_{U} (x)=− ln F_{U} (x)$ the cumulative reversed hazard rate function of a random variable U, we prove that both random variables T_Y(X) and T_X(Y) possess certain suitable positive ageing properties if and only if lifetimes X and Y have proportional reversed hazard rates.

Note that throughout the paper the terms ‘increasing’ and ‘decreasing’ are used in non-strict sense. Moreover, we denote by $H^{−1} (u)= sup {x : H(x)⩽u}$ the right-continuous inverse of a function H(x).

Section snippets

Proportional reversed hazards model

Let X be a random lifetime, i.e. a non-negative absolutely continuous random variable, with probability density function f_X(x), cumulative distribution function F_X(x) and survival function $F ̄_{X} (x)=1−F_{X} (x)$ . Assume that (0,b) is the interval of support of F_X(x), with 0<b⩽∞. The following definitions are well known (see Shaked and Shanthikumar (1994), for instance): $hazard rate function of X : r_{X} (x)=− d d x ln F ̄_{X} (x)= f_{X} (x) F ̄_{X} (x), x<b,$ $reversed hazard rate function of X : τ_{X} (x)= d d x ln F_{X} (x)= f_{X} (x) F_{X} (x), x>0.$ These functions

Preservation of ageing properties

Hereafter we list various well-known ageing notions in order to show some preservation results of ageing properties under the proportional reversed hazards model. A random lifetime X is said to be (see Shaked and Shanthikumar, 1994, for instance)

•
increasing in likelihood ratio (ILR) if f_X(x) is logconcave, i.e. if f_X(x+t)/f_X(x) is decreasing in x for all positive t;
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decreasing reversed failure rate (DRFR) if F_X(x) is logconcave, i.e. if τ_X(x) is decreasing in x;
•
increasing failure rate (IFR) if $F ̄_{X}$

Some properties based on stochastic orders

In this section we show some properties of the proportional reversed hazards model which are based on stochastic comparisons. To this purpose, let us recall some definitions of stochastic orders. Let X and Y be non-negative absolutely continuous random variables; X is said to be smaller than Y in the

•
likelihood ratio order (denoted by X⩽_lrY) if f_X(x)/f_Y(x) decreases in x;
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hazard rate order (denoted by X⩽_hrY) if $F ̄_{X} (x)/ F ̄_{Y} (x)$ decreases in x, i.e. if r_X(x)⩾r_Y(x) for all x;
•
reversed hazard rate order

Relative ageing of distributions

Let X and Y be two non-negative absolutely continuous random variables describing random lifetimes. In order to study the reliability of X relative to Y, recently Sengupta and Deshpande (1994) and Rowell and Siegrist (1998) have considered the following concepts on relative ageing (for a random lifetime U, here $R_{U} (x)=− ln F ̄_{U} (x)= ∫ 0 x r_{U} (u) d u$ denotes the cumulative hazard rate function of U):

•
X is said IFR (IFRA, NBU) relative to Y if R_Y(X) is IFR (IFRA, NBU) in the usual sense.

These three notions

Acknowledgments

The author thanks a referee and an associate editor for useful comments which improved the presentation of the paper. Moreover, he thanks Professors F. Pellerey and K. Siegrist for providing him with some references.

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Some results on the proportional reversed hazards model

Abstract

Introduction

Section snippets

Proportional reversed hazards model

Preservation of ageing properties

Some properties based on stochastic orders

Relative ageing of distributions

Acknowledgments

The reversed hazard rate function

Probab. Eng. Inf. Sci.

Imperfect repair

J. Appl. Probab.

Modeling failure time data by Lehman alternatives

Comm. Statist. Theory Methods