Nonparametric Estimation of the Bivariate Recurrence Time Distribution

This paper considers statistical models in which two different types of events, such as the diagnosis of a disease and the remission of the disease, occur alternately over time and are observed subject to right censoring. We propose nonparametric estimators for the joint distribution of bivariate recurrence times and the marginal distribution of the first recurrence time. In general, the marginal distribution of the second recurrence time cannot be estimated due to an identifiability problem, but a conditional distribution of the second recurrence time can be estimated non-parametrically. In literature, statistical methods have been developed to estimate the joint distribution of bivariate recurrence times based on data of the first pair of censored bivariate recurrence times. These methods are efficient in the current model because recurrence times of higher orders are not used. Asymptotic properties of the estimators are established. Numerical studies demonstrate the estimator performs well with practical sample sizes. We apply the proposed method to a Denmark psychiatric case register data set for illustration of the methods and theory

Joint Modeling and Estimation for Recurrent Event Processes and Failure Time Data

Recurrent event data are commonly encountered in longitudinal follow-up studies related to biomedical science, econometrics, reliability, and demography. In many studies, recurrent events serve as important measurements for evaluating disease progression, health deterioration, or insurance risk. When analyzing recurrent event data, an independent censoring condition is typically required for the construction of statistical methods. Nevertheless, in some situations, the terminating time for observing recurrent events could be correlated with the recurrent event process and, as a result, the assumption of independent censoring is violated. In this paper, we consider joint modeling of a recurrent event process and a failure time in which a common subject-specific latent variable is used to model the association between the intensity of the recurrent event process and the hazard of the failure time. The proposed joint model is flexible in that no parametric assumptions on the distributions of censoring times and latent variables are made and, under the model, informative censoring is allowed for observing both the recurrent events and failure times. We propose a ‘borrow-strength estimation procedure’ by first estimating the value of the latent variable from recurrent event data, and next using the estimated value in the failure time model. Some interesting implications and trajectories of the proposed model will be presented. Properties of the regression parameter estimates and the estimated baseline cumulative hazard functions are also studied