Empirical likelihood in single-index quantile regression with high dimensional and missing observations

Based on empirical likelihood method, we investigate statistical inference in partially linear single-index quantile regression with high dimensional linear and single-index parameters when the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. In particular, applying B-spline approximation to the unknown link function, we establish asymptotic normality of bias-corrected empirical likelihood ratio function and maximum empirical likelihood estimators of the parameters. Variable selection is considered by using the SCAD penalty. Meanwhile, we propose a penalized empirical likelihood ratio statistic to test hypothesis, and prove its asymptotically chi-square distribution under the null hypothesis. Also, simulation study and a real data analysis are conducted to evaluate the performance of the proposed methods.