Bandwidth selection for the estimation of transition probabilities in the location-scale progressive three-state model

Times between consecutive events are often of interest in medical studies. Usually the events represent different states of the disease process and are modeled using multi-state models. This paper introduces and studies a feasible estimation method for the transition probabilities in a progressive three-state model. We assume that the vector of gap times $(T_1,T_2)$ satisfies a nonparametric location-scale regression model $T_2=m(T_1)+\sigma (T_1)\epsilon $ , where the functions $m$ and $\sigma $ are ‘smooth’, and $\epsilon $ is independent of $T_1$ . Under this model, Van Keilegom et al. (J Stat Plan Inference 141:1118–1131, 2011) proposed estimators of the transition probabilities. However, the important issue of automatic bandwidth choice in this setting has not been examined, making the analysis of real datasets rather difficult. In this paper, we study the performance of their estimator in practice, we propose some modifications and study practical issues related to the implementation of the estimator, which involves the choice of an appropriate bandwidth. In an extensive simulation study the good performance of the method is shown. Simulations also demonstrate that the proposed estimator compares favorably with alternative estimators. Furthermore, the proposed methodology is illustrated with a real database on breast cancer.