Large Sample Theory for Semiparametric Regression Models with Two-Phase, Outcome Dependent Sampling

Outcome-dependent, two-phase sampling designs can dramatically reduce the costs of observational studies by judicious selection of the most informative subjects for purposes of detailed covariate measurement. Here we derive asymptotic information bounds and the form of the efficient score and influence functions for the semiparametric regression models studied by Lawless, Kalbfleisch, and Wild (1999) under two-phase sampling designs. We show that the maximum likelihood estimators for both the parametric and nonparametric parts of the model are asymptotically normal and efficient. The efficient influence function for the parametric part aggress with the more general information bound calculations of Robins, Hsieh, and Newey (1995). By verifying the conditions of Murphy and Van der Vaart (2000) for a least favorable parametric submodel, we provide asymptotic justification for statistical inference based on profile likelihood.

Locally Efficient Estimation in Censored Data Models: Theory and Examples

In many applications the observed data can be viewed as a censored high dimensional full data random variable X. By the curve of dimensionality it is typically not possible to construct estimators that are asymptotically efficient at every probability distribution in a semiparametric censored data model of such a high dimensional censored data structure. We provide a general method for construction of one-step estimators that are efficient at a chosen submodel of the full-data model, are still well behaved off this submodel and can be chosen to always improve on a given initial estimator. These one-step estimators rely on good estimators of the censoring mechanism and thus will require a parametric or semiparametric model for the censoring mechanism. We present a general theorem that provides a template for proving the desired asymptotic results. We illustrate the general one-step estimation methods by constructing locally efficient one-step estimators of marginal distributions and regression parameters with right-censored data, current status data and bivariate right-censored data, in all models allowing the presence of time-dependent covariates. The conditions of the asymptotics theorem are rigorously verified in one of the examples and the key condition of the general theorem is verified for all examples.