Dimension-reduced semiparametric estimation of distribution functions and quantiles with nonignorable nonresponse

Abstract

To estimate distribution functions and quantiles of a response variable when the data having nonignorable nonresponse and the dimension of covariate is not low, this article assumes that the propensity follows a general semiparametric model, but the distribution of the response variable and related covariates is unspecified. To address the identifiability problem, an instrumental covariate, which is related to the response variable but unrelated to the propensity given the response variable and other covariates, is used to construct sufficient instrumental estimating equations. Three different semiparametric estimation methods are developed based on inverse probability weighting, mean imputation, and augmented inverse probability weighting. Furthermore to improve the efficiency and alleviate the curse of dimensionality, the sufficient dimension reduction technique is employed to produce efficient kernel estimation, and a class of dimension-reduced estimators for distribution functions and quantiles is proposed. Consistency and asymptotic normality of the proposed estimators are established. It can be shown that these estimators are asymptotically equivalent. The finite-sample performance of the estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.