Detection of the symmetry of model errors for partial linear single-index models

In this paper, we propose a k-th correlation coefficient estimator between the density function and distribution function of the model errors in single-index models and partial linear single-models. This k-th correlation coefficient estimator is used to test whether the density function of the true model error is symmetric or not. First, we propose a moment based estimator of k-th correlation coefficient and present its asymptotic results. Second, we consider statistical inference of the k-th correlation coefficient estimator by using the empirical likelihood method. The empirical likelihood statistic is shown to be asymptotically distributed a centered chi-squared distribution with degree of freedom one. Simulation studies are conducted to examine the performance of the proposed estimators and test statistics.