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

Varying-coefficient model Y = ∑ j = 1 p β j ( U ) X j + ɛ 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 β ^ ( · ) is derived when the selection probabilities are known, estimated parametrically or nonparametrically. We show that the resulting nonparametric estimator of β ^ ( · ) 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.