Semiparametric Efficient Estimation of a Conditional Density with Missing or Mismeasured Covariates

Abstract

SUMMARY Pepe and Fleming, and Carroll and Wand have recently proposed estimators in a parametric model for the density of a random variable Y conditional on a vector of covariates (X, V) when data on one of the regressors X is missing for some study subjects. We propose a new class of estimators that remains consistent and asymptotically normal even when the probability that X is missing depends on the observed V and Y, includes an estimator whose asymptotic variance attains the semiparametric variance bound for the model and, when the data are missing completely at random, includes an estimator that is asymptotically equivalent to the inefficient estimators proposed by Pepe and Fleming and by Carroll and Wand. The optimal estimator in our class depends on the unknown probability law generating the data. When the vector V of non-missing regressors has at most two continuous components, we propose an adaptive semiparametric efficient estimator and compare the performance of the proposed semiparametric efficient estimator with the estimators proposed by Pepe and Fleming and Carroll and Wand in a small simulation study. When V has many continuous components, we propose an alternative class of adaptive estimators that should have high efficiency.