Quantiles and expectiles are determined by different loss functions: asymmetric least absolute deviation for quantiles and asymmetric squared loss for expectiles. This distinction ensures that quantile regression methods are robust to outliers but somewhat less effective than expectile regression, especially for normally distributed data. However, expectile regression is vulnerable to lack of robustness, especially for heavy-tailed distributions. To address this trade-off between robustness and effectiveness, we propose a novel approach. By introducing a parameter $\gamma$ that ranges between 0 and 1, we combine the aforementioned loss functions, resulting in a hybrid approach of quantiles and expectiles. This fusion leads to the estimation of a new type of location parameter family within the linear regression framework, termed Hybrid of Quantile and Expectile Regression (HQER). The asymptotic properties of the resulting estimaror are then established. Through simulation studies, we compare the asymptotic relative efficiency of the HQER estimator with its competitors, namely the quantile, expectile, and $k$th power expectile regression estimators. Our results show that HQER outperforms its competitors in several simulation scenarios. In addition, we apply HQER to a real dataset to illustrate its practical utility.

翻译：分位数与期望分位数由不同的损失函数确定：分位数采用非对称最小绝对偏差损失，而期望分位数采用非对称平方损失。这一区别确保了分位数回归方法对异常值具有稳健性，但其效率通常低于期望分位数回归，尤其在数据服从正态分布时更为明显。然而，期望分位数回归缺乏稳健性，尤其在处理重尾分布时表现脆弱。为权衡稳健性与效率之间的平衡，本文提出了一种新颖的方法。通过引入一个取值范围在0到1之间的参数$\gamma$，我们将上述两种损失函数相结合，从而形成了一种分位数与期望分位数的混合方法。这种融合在线性回归框架内推导出一类新型位置参数族的估计，称为分位数与期望分位数混合回归（HQER）。我们随后建立了所得估计量的渐近性质。通过模拟研究，我们比较了HQER估计量与其竞争方法（包括分位数回归、期望分位数回归以及$k$次幂期望分位数回归估计量）的渐近相对效率。结果表明，在多种模拟情境下，HQER均优于其竞争方法。此外，我们将HQER应用于实际数据集，以展示其实际应用价值。