Semiparametric Smoothed Quantile Regression with Nonignorable Nonresponse Based on Sufficient Dimension Reduction
This paper examines the problem of estimation in a quantile regression model when responses are missing not at random(MNAR).Firstly,we build a semi-parametric exponential tilting response model.In order to overcome challenge on identify and curse of dimensionality caused by multivariate nonparametric kernel es-timation,data-driven method based on assumption of sufficient dimension reduction is applied to construct nonresponse variables.profile two-step GMM estimator for tilt-ing parameters and dimension-reduced kernel estimator for nonparametric funtions are obtained.Then three kinds of unbiasd quantile regression estimation equations,namely inverse probability weighted(IPW),kernel assisted estimation equation im-putation(EEI)and augmented inverse probability weighted(AIPW)are established.quantile loss functions are replaced by convolution-type smoothed countparts which can avoid theoretical and computational difficulties caused by unsmoothness of check function.Empirical likelihood is employed to estimate quantile regression coefficients.The asymptotic normality of the three estimators and the asymptotic x2 weighting properties of the corresponding logarithmic empirical likelihood ratio functions are proved by theoretical studies.Simulation studies are provided to evaluate the finite sample performance of proposed method.Finally,the analysis of the data set of HIV-CD4 is presented.
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