Empirical likelihood for partially linear single-index models with missing observations

In this paper, we study the empirical likelihood for a partially linear single-index model with a subset of covariates and response missing at random. By using the bias correction and the imputation method, two empirical log-likelihood ratios are proposed such that any of two ratios is asymptotically chi-squared. Two maximum empirical likelihood estimates of the index coefficients and the estimator of link function are constructed, their asymptotic distributions and optimal convergence rate are obtained. It is proved that our methods yield asymptotically equivalent estimators for the index coefficients. An important feature of our methods is their ability to handle missing response and/or partially missing covariates. In addition, we study the estimation and empirical likelihood for two special cases the single-index model and partially linear model with observations are missing at random. A simulation study indicates that the proposed methods are comparable for bias and standard deviation, as well as in terms of coverage probabilities and average areas (lengths) of confidence regions (intervals). The proposed methods are illustrated by an example of real data. (C) 2019 Elsevier B.V. All rights reserved.