On competing risk and degradation processes

Lehmann’s ideas on concepts of dependence have had a profound effect on mathematical theory of reliability. The aim of this paper is two-fold. The first is to show how the notion of a “hazard potential” can provide an explanation for the cause of dependence between life-times. The second is to propose a general framework under which two currently discussed issues in reliability and in survival analysis involving interdependent stochastic processes, can be meaningfully addressed via the notion of a hazard potential. The first issue pertains to the failure of an item in a dynamic setting under multiple interdependent risks. The second pertains to assessing an item’s life length in the presence of observable surrogates or markers. Here again the setting is dynamic and the role of the marker is akin to that of a leading indicator in multiple time series. 1. Preamble: Impact of Lehmann’s work on reliability Erich Lehmann’s work on non-parametrics has had a conceptual impact on reliability and life-testing. Here two commonly encountered themes, one of which bears his name, encapsulate the essence of the impact. These are: the notion of a Lehmann Alternative, and his exposition on Concepts of Dependence. The former (see Lehmann [4]) comes into play in the context of accelerated life testing, wherein a Lehmann alternative is essentially a model for accelerating failure. The latter (see Lehmann [5]) has spawned a large body of literature pertaining to the reliability of complex systems with interdependent component lifetimes. Lehmann’s original ideas on characterizing the nature of dependence has helped us better articulate the effect of failures that are causal or cascading, and the consequences of lifetimes that exhibit a negative correlation. The aim of this paper is to propose a framework that has been inspired by (though not directly related to) Lehmann’s work on dependence. The point of view that we adopt here is “dynamic”, in the sense that what is of relevance are dependent stochastic processes. We focus on two scenarios, one pertaining to competing risks, a topic of interest in survival analysis, and the other pertaining to degradation and its markers, a topic of interest to those working in reliability. To set the stage for our development we start with an overview of the notion of a hazard potential, an entity which helps us better conceptualize the process of failure and the cause of interdependent lifetimes. ∗Research supported by Grant DAAD 19-02-01-0195, The U. S. Army Research Office. Department of Statistics, The George Washington University, Washington, DC 20052, USA, e-mail: [email protected] AMS 2000 subject classifications: primary 62N05, 62M05; secondary 60J65.