On locally weighted estimation and hypothesis testing of varying-coefficient models with missing covariates

Varying-coefficient model Y = Sigma(p)(j=1) beta(j)(U)X-j+epsilon has been studied extensively when data are completely observed. When the covariates X are missing at random, we propose a locally weighted estimator based on the inverse selection probabilities. Distribution theory of (beta) over cap(center dot) is derived when the selection probabilities are known, estimated parametrically or nonparametrically. We show that the resulting nonparametric estimator of (beta) over cap(center dot) when the selection probabilities are estimated nonparametrically has a smaller asymptotic variance than that when the selection probabilities are known or estimated parametrically. Motivated by Robin et al. [1994. Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89, 846-866], we also consider simple locally augmented weighted estimator. However, we show that it does not improve the efficiency theoretically. We have constructed a bootstrap test for goodness of fit of models in the missing covariates case. The results of a simulation study are also given to illustrate our method. The proposed method is applied to analyze an AIDS dataset from a clinical study. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.