Smooth goodness-of-fit tests for composite hypothesis in hazard based models

• 成果类型: Article
• 署名作者: Peña, EA
• 署名单位: University System of Ohio; Bowling Green State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1998
• 页码: 1935-1971
• 关键词: randomly censored-data; data-driven version; life-history data; counting-processes; regression-models; RESIDUALS; parameters; statistics; inference; sample

摘要

Consider a counting process (N(t), t epsilon T) with compensator process (A(t), t epsilon T), where A(t) = integral(o)(t)Y(s)lambda(o)(s) ds, (Y(t), t epsilon T) is an observable predictable process, and lambda(0)((.)) is an unknown hazard rate function. A general procedure for extending Neyman's smooth goodness-of-bt test for the composite null hypothesis H-o: lambda(0)((.)) epsilon C = (lambda(0)((.); eta): eta epsilon Gamma subset of or equal to R-q) is proposed and developed. The extension is obtained by embedding C in the class A(k) whose members are of the form lambda(0)((.);eta)exp(theta(t)psi((.); eta)), (eta, theta) epsilon Gamma x R-k, where psi((.); eta) = (psi(1)((.); eta),..., psi(k)((.); eta))(t) is a vector of observable random processes satisfying certain regularity conditions. The tests are based on quadratic forms of the statistic integral(0)(tau)(s; <(eta)over cap>)dM(s; <(eta)over cap>), where M(t; eta) = N(t) - integral(0)(t)Y(s)lambda(0)(s; eta) ds and <(eta)over cap> is a restricted maximum likelihood estimator of eta. Asymptotic properties of the test statistics are obtained under a sequence of local alternatives, and the asymptotic local powers of the tests are examined. The effect of estimating eta by <(eta)over cap> is ascertained, and the problem of choosing the psi-process is discussed. The procedure is illustrated by developing tests for testing that lambda(0)((.)) belongs to (i) the class of constant hazard rates and (ii) the class of Weibull hazard rates, with particular emphasis on the random censorship model. Simulation results concerning the achieved levels and powers of the tests are presented, and the procedures are applied to three data sets that have been considered in the literature.