Semiparametric Analysis of General Additive-Multiplicative Hazard Models for Counting Processes

Abstract

The additive-multiplicative hazard model specifies that the hazard function for the counting process associated with a multidimensional covariate process $Z = (W^T, X^T)^T$ takes the form of $\lambda(t\mid Z) = g\{\beta^T_0 W(t)\} + \lambda_0(t)h\{\gamma^T_0X(t)\}$, where $\theta_0 = (\beta^T_0, \gamma^T_0)^T$ is a vector of unknown regression parameters, $g$ and $h$ are known link functions and $\lambda_0$ is an unspecified "baseline hazard function." In this paper, we develop a class of simple estimating functions for $\theta_0$, which contains the partial likelihood score function in the special case of proportional hazards models. The resulting estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions. Weak convergence of the Aalen-Breslow type estimators for the cumulative baseline hazard function $\Lambda_0(t) = \int^t_0\lambda_0(u) du$ is also established. Furthermore, we construct adaptive estimators for $\theta_0$ and $\Lambda_0$ that achieve the (semiparametric) information bounds. Finally, a real example is provided along with some simulation results.