ESTIMATION OF REGRESSION-COEFFICIENTS WHEN SOME REGRESSORS ARE NOT ALWAYS OBSERVED

In applied problems it is common to specify a model for the conditiona l mean of a response given a set of regressors. A subset of the regres sors may be missing for some study subjects either by design or happen stance. In this article we propose anew class of semiparametric estima tors, based on inverse probability weighted estimating equations, that are consistent for parameter vector alpha(0) of the conditional mean model when the data are missing at random in the sense of Rubin and th e missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a re presentation for the efficient score, the semiparametric variance boun d, and the influence function of any regular, asymptotically linear es timator in this more general estimation problem. Because the optimal e stimator depends on the unknown probability law generating the data, w e propose locally and globally adaptive semiparametric efficient estim ators. We compare estimators in our class with previously proposed est imators. We show that each previous estimator is asymptotically equiva lent to some, usually inefficient, estimator in our class. This equiva lence is a consequence of a proposition stating that every regular asy mptotic linear estimator of alpha(0) is asymptotically equivalent to s ome estimator in our class. We compare various estimators in a small s imulation study and offer some practical recommendations.