Pseudo maximum likelihood estimation for the Cox model with doubly truncated data

The partial likelihood (PL) function has been mainly used for the Cox proportional hazards models with censored data. The PL approach can also be used for analyzing left-truncated or left-truncated and right-censored data. However, when data is subject to double truncation, the PL approach no longer works due to the complexities of risk sets. In this article, we propose pseudo maximum likelihood approach for estimating regression coefficients and baseline hazard function for the Cox model with doubly truncated data. We propose expectation-maximization algorithms for obtaining the pseudo maximum likelihood estimators (PMLE). The consistency property of the PMLE is established. Simulations are performed to evaluate the finite-sample performance of the PMLE. The proposed method is illustrated using an AIDS data set.