The Efficiency Bound of the Mixed Proportional Hazard Model

Abstract

The semiparametric efficiency bound of the mixed proportional hazard model is derived. The density of the model factors in such a way that there exists a complete sufficient statistic for the individual heterogeneity. The efficient score is shown to be the difference between the score in the parametric direction and its conditional expectation given the sufficient statistic. Applying this result to the single-spell Weibull mixed proportional hazard model, it is shown that its information matrix is singular and there cannot exist any ^/w-consistent estimator sequence. The information of the multi-spell Weibull mixed proportional hazard model is shown to be nonsingular in general.