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In this paper, we propose a residual-based estimator of  k -th correlation coefficient between the density function and distribution function of a continuous variable for partial linear regression models, and further we use this  k -th correlation coefficient 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  k -th correlation coefficient by using the empirical likelihood method, and the empirical likelihood statistic is shown to be asymptotically distributed as Chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators.

Detection the symmetry or asymmetry of model errors in partial linear models