Aging properties of the additive and proportional hazard mixing models
Introduction
The fact that any device or system shows an increasing failure rate is reasonably due to age or use, or both may cause the mechanism to wear out over time. Nevertheless, a decreasing failure rate (DFR) that means an improvement as time goes by, is far from being easy to explain, and the reasons for it being, in general, less intuitive.
Mixtures of lifetime distributions turn out to be the most widespread explanation for this ‘positive aging’. Proschan [20], found that a mixture of exponential distributions was the appropriate choice to model the failures in the air-conditioning systems of planes. Such mixing was the reason of the DFR that the aggregated data exhibited. DFRs also arise as a consequence of burn-in [6], [18].
The effect of mixtures of distributions on its aging characteristics (reliability function, failure rate, failure rate average, and mean residual life) has been widely studied. Many works focus on preserving properties: Barlow and Proschan [4], Brown [8] and Klefsjö [17] prove, respectively, the preservation under mixtures of DFR, increasing mean residual life (IMRL), and the harmonic new worse than used in expectation (HNWUE) classes. Other works analyze the shape of the hazard rate of a mixture: Block [5], explains some surprising results involving the hazard rate of mixtures of lifetime distributions and discuss its asymptotic behavior. Finkelstein and Esaulova [10], study the limiting behavior of the mixture failure rate when the mixture consists of distributions under the proportional hazards model. They show that conditional expectation and variance are the key parameters determining the tendency of the mixture failure rate. The tail behavior of the failure rate function of mixtures has been also studied by Block and Joe [7]. Badı́a et al. [2] give bounds for the derivatives of the aging characteristics in mixtures.
The conditions under which systems operate can be harsher or gentler and mixtures also arise when modeling lifetime of systems in a changing environment. The Cox model [9], where the changing conditions are assumed to act multiplicatively on the baseline failure rate, is the most known mixture model. Gupta and Gupta [16], study the relation between the conditional and unconditional failure rates in mixtures when the distributions in the mixture follow the proportional hazard rate. However, when the difference of risks under two different conditions is considered, the additive risk model [1], [18], [19] should be used.
In a recent work, Finkelstein and Esaulova [11] deal with what they called ‘the inverse problem’, that is, given the mixture failure rate and the mixing distribution, obtain the failure rate of the baseline distribution. This operation is carried out for the two models of mixing with additive and proportional hazard rates. Moreover, Finkelstein and Esaulova [12] analyze the limiting behavior of the failure rate in both types of mixtures and study how the increasing failure rate is reversed as t→∞. Finkelstein [13] obtain conditions under which the failure rate and the reciprocal to the mean residual life exhibit equivalent asymptotic behavior. In addition, Finkelstein [13] describe the mean residual life of mixtures as wmoveNomenell as some asymptotic results on the shape of the mean residual life corresponding to the direct proportional model mixtures.
In this work, mixtures of lifetime distributions following the additive risk and the proportional hazard models are studied under a different approach than in the foregoing references. In both cases, we provide the relations between the hazard rate and hazard rate average of the mixture with that corresponding to an exponential mixture. It is important to note that properties of stochastic aging should be taken into account when modeling times to failure. Thus, we aim at answering how the mixture and the distributions that compound it are connected: the properties of the mixing which are induced by the pattern of aging of the distributions in the mixture and, conversely, the properties of the baseline distribution that may be derived from the aging characteristics of the mixture.
Section snippets
The additive hazard model
The additive hazard model considers a baseline hazard rate, r(x), corresponding to a non-negative distribution and a random variable, Z, representing the changes in the operating conditions with an additive effect on r(x). This means that the conditional failure rate, given Z=z isThe expression in Eq. (2.1) can result as a consequence of the effect of a random environment on a failure rate r(x).
In what follows, Z that describes heterogeneous populations or operating conditions, is
The proportional hazard model
The proportional hazard model due to Cox [9], has been widely used in many experiments where the time to failure depends on a group of covariates. These covariates may represent different treatments, operating conditions, heterogeneous environments, etc.
The model considers a baseline function, r(x), which is the failure rate of a non-negative random variable. The changing factors, represented by Z, have in this case a multiplicative effect on r(x). Therefore, the failure rate of the system,
Examples
Proposition 1, Proposition 2, as well as Theorem 1, Theorem 2 are worthwhile results so as to study the shape of the hazard rate and hazard rate average of a mixture. As a matter of fact, the expressions of the aging characteristics are usually hard to obtain and the characterizations in Proposition 1, Proposition 2 simplify the calculations. Theorem 1, Theorem 2 constitute a remarkable help to know the way the mixture behaves.
In the following examples we will consider three different types of
Acknowledgements
Thanks are due to the Editor as well as to Max Finkelstein and an anonymous referee for their helpful comments that improved the writing of this article. The authors also thank César Berrade for its valuable assistance.
References (20)
- et al.
Why the mixture failure rate decreases
Reliab Engng Syst Saf
(2001) A model for nonparametric regression analysis of counting processes
- et al.
On the behavior of aging characteristics in mixed populations
Probab Engng Inform Sci
(2001) - Badı́a FG, Berrade MD, Campos CA. Proportional mean residual life model mixtures. Preprint;...
- et al.
Statistical theory of reliability and life testing
(1981) The shape of the hazard rate of a mixture
- et al.
Burn-in
Stat Sci
(1997) - et al.
Tail behaviour of the failure rate functions of mixtures
Lifetime Data Anal
(1997) Further monotonicity properties for specialized renewal processes
Ann Probab
(1981)Regression models and life tables (with discussion)
J R Stat Soc
(1972)
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