Semiparametric transformation models with random effects for recurrent events

In this article we study a class of semiparametric transformation models with random effects for the intensity function of the counting process. These models provide considerable flexibility in formulating the effects of possibly time-dependent covariates on the developments of recurrent events while accounting for the dependence of the recurrent event times within the same subject. We show that the nonparametric maximum likelihood estimators (NPMLEs) for the parameters of these models are consistent and asymptotically normal. The limiting covariance matrices for the estimators of the regression parameters achieve the semiparametric efficiency bounds and can be consistently estimated. The limiting covariance function for the estimator of any smooth functional of the cumulative intensity function also can be consistently estimated. We develop a simple and stable EM algorithm to compute the NPMLEs as well as the variance and covariance estimators. Simulation studies demonstrate that the proposed methods perform well in practical situations. Two medical studies are provided for illustrations.