Stochastic Justification of Some Simple Reliability Models∗

In this paper we derive conditions on the internal wear process under which the resulting time-to-failure model will be of the simple collapsible form (Oakes, 1995, Duchesne and Lawless, 2000) when the usage accumulation history is available. We suppose that failure occurs when internal wear crosses a certain threshold and/or a traumatic event causes the item to fail (Cox, 1999 and Bagdonavičius and Nikulin, 2001). We model the infinitesimal increment in internal wear as a function of time, accumulated internal wear and usage history, and we derive conditions on this function to get a collapsible model for the distribution of time-to-failure given the usage history. We reach the conclusion that collapsible models form the subset of accelerated failure time models with time-varying covariates (Robins and Tsiatis, 1992) for which the time transformation function satisfies certain simple properties.