Goodness-of-fit of the distribution of time-to-first-occurrence in recurrent event models.

Imperfect repair models are a class of stochastic models that deal with recurrent phenomena. This article focuses on the Block, Borges, and Savits (1985) age-dependent minimal repair model (the BBS model) in which a system that fails at time t undergoes one of two types of repair: with probability p(t), a perfect repair is performed, or with probability 1-p(t), a minimal repair is performed. The goodness-of-fit problem of interest concerns the initial distribution of the failure ages. In particular, interest is on testing the null hypothesis that the hazard rate function of the time-to-first-event-occurrence, lambda(.), is equal to a prespecified hazard rate function lambda 0(.). This paper extends the class of hazard-based smooth goodness-of-fit tests introduced in Peña (1998a) to the case where data accrual is from a BBS model. The goodness-of-fit tests are score tests derived by reformulating Neyman's idea of smooth tests in terms of hazard functions. Omnibus as well as directional tests are developed and simulation results are presented to illustrate the sensitivities of the proposed tests for certain types of alternatives.