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.