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Article Excerpt
1. Introduction
The idea of competing risks is to model the situation where units are exposed to several risks and fail due to one of them. We observe two random variables for each individual, (Y, [delta]), where Y is the time to failure of the individual, and 8 is the cause of failure.
Our motivation and main application in the present paper is the competing risks situation occurring when a potential component failure at some time X may be avoided by a Preventive Maintenance (PM) at time Z. The experienced event will in this case be at time Y = min(X, Z), and it will either be a failure or a PM. It is convenient to define [delta] = I(Z < X), where 1(A) is the indicator function of the event A. Thus, [delta] = means that the component fails and [delta] = 1 means that it is preventively maintained.
Note that the observable result is the pair (Y, [delta]), rather than the underlying times X and Z, which are often the times of interest. For example, knowing the distribution of X would be important as a basis for maintenance optimization. It is well known (Crowder, 2001, ch. 7), however, that in a competing risks case as described here, the marginal distributions of X and Z are not identifiable from observation of (Y, [delta]) alone unless specific assumptions are made on the dependence between X and Z. One such assumption is that X and Z are independent (Crowder, 2001, ch. 7). This assumption is not reasonable in our application, however, since the maintenance crew is likely to have some information regarding the component's state during operation. This insight is used to perform maintenance with the aim of avoiding component failures. We are thus in practice usually faced with a situation of dependent competing risks between X and Z.
Cooke (1993, 1996a, 1996b) introduced the notion of random signs censoring which is tailored for such cases. In our notation, random signs censoring can be defined as follows:
Definition 1. Let (X, Z) be a pair of life variables. Then Z is called a random signs censoring of X if the event {Z < X} is stochastically independent of X.
Thus, random signs censoring means that the event that the failure of the component is preceded by PM, is not influenced by the time X at which the component fails or would have failed without PM. The idea is that the component emits some kind of signal before failure, and that this signal is discovered with a probability which does not depend on the age of the component. Moreover, random signs censoring implies identiliability of the distribution of X, while the distribution of Z is not identifiable in general under these assumptions.
Lindqvist et at. (2006) suggested a model called the repair alert model for describing the joint behavior of failure times X and PM-times Z. This model is a special case of random signs censoring obtained by introducing a repair alert function which describes the "alertness" of the maintenance crew as a function of time.
In the present paper we suggest another modeling approach which again leads to a model satisfying the random signs property. The approach is based on modeling of the degradation of a component by means of Wiener processes, with failure corresponding to the first passage time of a certain level. As is well known (Chhikara and Folks, 1989), this implies that the failure time has an inverse Gaussian distribution. We extend the model by introducing another level, below the failure level, to represent PM. The PM may be performed when the degradation process reaches this level. Whitmore (1986) studied a similar case using first passage times of multi-dimensional Wiener processes to model independent competing risks. Horrocks and Thompson (2004) considered a Wiener process model for health status for hospitalized patients with two barriers, corresponding to discharge and death, respectively. Most interestingly they suggested an extension where a decision to transfer the patient is considered when the process reaches a certain intermediate level. This corresponds exactly to the possibility of performing PM before failure in our models. The approach of the present paper is, furthermore, related to the one of Whitmore et al. (1998) who considered a marker process which is correlated to the latent failure process. Aalen and Gjessing (2001) gave a review of failure time models based on first passage times of stochastic processes. For an application of inverse Gaussian distributions in accelerated life testing we refer to Doksum and Hoyland (1992).
The plan of the paper is as follows. In Section 2 we introduce some basic notation for competing risks. Section 3 reviews the concepts of random signs censoring and the repair alert model, which are examples of approaches for modeling failure versus PM. A short introduction to the Wiener process and the inverse Gaussian distribution is then given in Section 4, while in Sections 5 and 6 we present the new models and derive the likelihood functions which are needed for their analyses. A numerical study using a classical data set is presented in Section 7, where the result is compared to the result obtained using the exponential repair alert...
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