Untitled

Abstract

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.

A Smooth Test of Goodness-of-Fit for the Baseline Hazard Function for Time-to-First Occurrence in Recurrent Events: An Application to HIV Retention Data

“…= 10 and ! = 5," = 15 for the Weibull initial distribution and found the results to be consistent with those presented by [18][19][20]25]. Examining the performance of the directional tests, we again noticed that we are able to achieve required significance levels.…”

“…where k is some fixed positive integer and ! (") is a k !1 vector of locally bounded predictor process [12,[18][19][20]23]. Note that, the null hypothesis H 0 : !…”

“…(") , polynomial specification ( k ), smoothing process of ! (") , achieved power of S(t) and asymptotic distribution of S(t) , see [3,12,[17][18][19][20]23].…”

“…We conducted Monte Carlo simulations to investigate the performance of the smooth tests. Similar tests have been examined in extensive simulation studies [18][19][20]. The goal of simulations was to compare the empirical significance levels of the tests with the specified nominal asymptotic levels as the sample size and the degree of censoring are varied.…”

“…A major contribution of this paper is the application of the BBS model to a primary HIV retention dataset and the assessment of the model using the smooth test of goodness-of-fit. In modelling the LTFU data, we utilize stochastic formulation, which allows a direct estimation of nuisance parameters [19]. By fitting LTFU data to a parametric family of baseline hazard functions, we are able to make more efficient inferences about the risk.…”