Mixed Proportional Hazard Model Research Articles

Maximum penalized likelihood estimation of mixed proportional hazard models

Unobservable individual effects in models of duration will cause estimation bias that include the structural parameters as well as the duration dependence. The maximum penalized likelihood estimator is examined as an estimator for the survivor model with heterogeneity. Proofs of the existence and uniqueness of the maximum penalized likelihood estimator in duration model with general forms of unobserved heterogeneity are provided. Some small sample evidence on the behavior of the maximum penalized likelihood estimator is given. The maximum penalized likelihood estimator is shown to be computationally feasible and to provide reasonable estimates in most cases.

The Identifiability of the Mixed Proportional Hazards Model with Time-Varying Coefficients

This paper establishes conditions for the nonparametric identifiability of the mixed proportional hazards model with time-varying coefficients. Unlike the mixed proportional hazards model, a regressor with two distinct values is not sufficient to identify this model. An unbounded regressor, however, is sufficient for identification.

Constructive identification of the mixed proportional hazards model

We give a new proof of the identifiably of the MPH model. This proof is constructive: it is a recipe for constructing the triple—regression function, base‐line hazard, and distribution of the individual effect—from the observed cumulative distribution functions.We then prove that the triples do not depend continuously on the observed cumulative distribution functions. Uniformly consistent estimators do not exist.Finally we show that the MPH model is even identifiable from two‐sided censored observations. This proof is constructive, too.

The Efficiency Bound of the Mixed Proportional Hazard Model

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 '/ -consistent estimator sequence. The information of the multi-spell Weibull mixed proportional hazard model is shown to be nonsingular in general.

The Non-Parametric Identification of Generalized Accelerated Failure-Time Models

We consider a class of models that generalizes the popular Mixed Proportional Hazard (MPH) model for duration data: the Generalized Accelerated Failure-Time (GAFT) model. We show that the GAFT model is non-parametrically identified (up to a normalization). We then reconsider the non-parametric identification of the MPH model. We show that the class of MPH models is not closed under normalization. This implies that a finite mean of the mixing distribution is a necessary condition for (non-parametric) identification of the MPH model. It is impossible to test this hypothesis without imposing arbitrary restrictions on the base-line hazard and/or the regression function.