We study the varying-coefficient partially linear model when some linear covariates are not observed, but their auxiliary instrumental variables are available. Combining the calibrated error-prone covariates and modal regression, we present a two-stage efficient estimation procedure, which is robust against outliers or heavy-tail error distributions. Asymptotic properties of the resulting estimators are established. Performance of our proposed estimation procedure is illustrated through some numerous simulations and a real example. And the results confirm that the proposed methods are satisfactory.

We have proposed a robust and efficient variable selection method for ultrahigh-dimensional linear models with nonrandomly missing responses, leveraging modal regression. The propensity score function was specified by a semiparametric model and we introduced a two-step estimation procedure. In the first feature screening stage, the Pearson chi-square (PC) test statistic identifies significant predictors in the sparse propensity score model. The generalized method of moment (GMM) estimates parameters to obtain consistent estimation for the propensity score in the second stage. With the estimated propensity score, we suggested a feature screening and variable selection procedure based on the inverse probability weighting (IPW). A modified sure independence screening (SIS) method first reduces the model dimensionality, followed by a penalized modal regression approach to select significant covariates. The proposed procedure can deal with the ultrahigh-dimensional data with nonignorable nonresponse, and this modal-based procedure is robust against outliers and heavy-tailed errors. Additionally, we established the asymptotic properties of the estimators under mild regularity conditions. Simulation studies and real data applications confirm the method's effectiveness in finite samples and practical settings.