A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model

Abstract

Ridder and Woutersen (Ridder, G., and T. Woutersen. 2003. “The Singularity of the Efficiency Bound of the Mixed Proportional Hazard Model.” Econometrica 71: 1579–1589) have shown that under a weak condition on the baseline hazard, there exist root-N consistent estimators of the parameters in a semiparametric Mixed Proportional Hazard model with a parametric baseline hazard and unspecified distribution of the unobserved heterogeneity. We extend the linear rank estimator (LRE) of Tsiatis (Tsiatis, A. A. 1990. “Estimating Regression Parameters using Linear Rank Tests for Censored Data.” Annals of Statistics 18: 354–372) and Robins and Tsiatis (Robins, J. M., and A. A. Tsiatis. 1992. “Semiparametric Estimation of an Accelerated Failure Time Model with Time-Dependent Covariates.” Biometrika 79: 311–319) to this class of models. The optimal LRE is a two-step estimator. We propose a simple one-step estimator that is close to optimal if there is no unobserved heterogeneity. The efficiency gain associated with the optimal LRE increases with the degree of unobserved heterogeneity. Keywords: counting process; linear rank estimation; mixed proportional hazard