Nonparametric Sieve Maximum Likelihood Estimation of Semi-Competing Risks Data

In biomedical studies involving time-to-event data, a subject may experience distinct types of events. We consider the problem of estimating the transition functions for a semi-competing risks model under illness-death model framework. We propose to estimate the intensity functions by maximizing a B-spline based sieve likelihood. The method yields smooth estimates without parametric assumptions. Our proposed approach facilitates easy computation of the covariance of the model parameters and yields direct interpretation. Compared with existing approaches, our proposed method requires neither the subjective specification of the frailty distribution nor the Markov or semi-Markov assumption which may be unmet in real applications. We establish the consistency, the convergence rate, and the asymptotic normality of the proposed estimators under some regularity conditions. We also provide simulation studies to assess the finite-sample performance of the proposed modeling and estimation strategy. A real data application is further used to illustrate the proposed methodology.