Empirical likelihood in single-index partially functional linear model with missing observations

In this paper, we focus on the empirical likelihood in the single-index partially functional linear model. When the response variables or/and part of the covariates are missing at random, we construct the empirical likelihood ratio of the parameter in the model based on B-spline approximation for the link and slope functions, and define maximum empirical likelihood (MEL) estimator of the parameter. Under suitable assumptions, the asymptotic distributions of the proposed empirical log-likelihood ratio and MEL estimator are established. At the same time, based on penalized empirical likelihood (PEL) approach, we define the PEL estimator of the parameter and investigate variable selection of the model. A simulation study is done to evaluate the finite sample performance for the proposed methods.