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

Abstract

Consider a counting process $N(t), t\inT}$ with compensator process ${A(t),t\in T}$, where $A(t)=\int_0^t Y(s) ds, {Y(t), t\in T}$ is an observable predictable process, and $\lambda_0(\dot)$ is an unknown hazard rate function. A general procedure for extending Neyman’s smooth goodness­of­fit test for the composite null hypothesis $H_0: \lambda_0(\dot)\inC ={\lambda_0(\dot;\eta):\eta\in\Gamma\subseteq\Re^q}$ is proposed and developed. The extension is obtained by embedding $C$ in the class $A_ k$ whose members are of the form $\lambda_0(\dot;\eta)\exp{\theta^t\psi(\dot;\eta)}, (\eta,\theta) \in\Gamma\times\Re^k$, where $\psi(\dot;\eta)=(\psi_1(\dot;\eta,\ldots,\psi_k(\dot;\eta))^t$ is a vector of observable random processes satisfying certain regularity conditions. The tests are based on quadratic forms of the statistic $\int_0^\tau\psi(s;\hat{\eta})dM(s;\hat{\eta})$, where $M(t;\eta) = N(t) - \int_0^t Y(s)\lambda_0(s;\eta) ds$ and $\hat {\eta}$ 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 $\hat{\eta}$ is ascertained, and the problem of choosing the $\lambda$­process is discussed. The procedure is illustrated by developing tests for testing that $\lambda_0(\dot)$ 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.