Statistica Sinica 13(2003), 519-537

EFFICIENT ESTIMATION FOR THE PROPORTIONAL

HAZARDS MODEL WITH LEFT-TRUNCATED AND

``CASE 1'' INTERVAL-CENSORED DATA

Jong S. Kim

Portland State University

Abstract: The maximum likelihood estimator (MLE) for the proportional hazards model with left-truncated and ``Case $1$'' interval-censored data is studied. Under appropriate regularity conditions, the MLE of the regression parameter is shown to be asymptotically normal with a root-n convergence rate and achieves the information bound, even though the difference between left-truncation time and censoring time of the MLE of the baseline cumulative hazard function converges only at rate $n^{1/3}$. Two methods to estimate the variance-covariance matrix of the MLE of the regression parameter are considered. One is based on a generalized missing information principle and the other is based on the profile information procedure. Simulation studies show that both methods work well in terms of bias and variance for samples of moderate sizes. An example is provided to illustrate the methods.

Key words and phrases: Asymptotic distribution, left-truncated and ``Case 1'' interval-censored data, proportional hazards model, variance estimation.