The kth Power Expectile Estimation and Testing

This paper develops the theory of the kth power expectile estimation and considers its relevant hypothesis tests for coefficients of linear regression models. We prove that the asymptotic covariance matrix of kth power expectile regression converges to that of quantile regression as k converges to one and hence promise a moment estimator of asymptotic matrix of quantile regression. The kth power expectile regression is then utilized to test for homoskedasticity and conditional symmetry of the data. Detailed comparisons of the local power among the kth power expectile regression tests, the quantile regression test, and the expectile regression test have been provided. When the underlying distribution is not standard normal, results show that the optimal k are often larger than 1 and smaller than 2, which suggests the general kth power expectile regression is necessary. Finally, the methods are illustrated by a real example.