Check functions of least absolute deviation make sure quantile regression methods are robust, while squared check functions make expectiles more sensitive to the tails of distributions and more effective for the normal case than quantiles. In order to balance robustness and effectiveness, we adopt a loss function, which falls in between the above two loss functions, to introduce a new kind of expectiles and develop an asymmetric least k th power estimation method that we call the k th power expectile regression, k larger than 1 and not larger than 2. The asymptotic properties of the corresponding estimators are provided. Simulation results show that the asymptotic efficiency of the k th power expectile regression is higher than those of the common quantile regression and expectile regression in some data cases. A primary procedure of choosing satisfactory k is presented. We finally apply our method to the real data.

中文翻译：

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第 k 次幂期望回归

最小绝对偏差的检查函数确保分位数回归方法是稳健的，而平方检查函数使期望值对分布的尾部更敏感，对正常情况比分位数更有效。为了平衡鲁棒性和有效性，我们采用了一个介于上述两个损失函数之间的损失函数，引入了一种新的期望值，并开发了一种非对称的最小 k 次幂估计方法，我们称之为 k 次幂期望回归, k 大于 1 且不大于 2。提供了相应估计量的渐近特性。仿真结果表明，在某些数据情况下，k次方期望回归的渐近效率高于普通分位数回归和期望回归。介绍了选择满意 k 的主要程序。我们最终将我们的方法应用于真实数据。