Expectile regression is a useful tool for exploring the relation between the response and the explanatory variables beyond the conditional mean. A continuous threshold expectile regression is developed for modeling data in which the effect of a covariate on the response variable is linear but varies below and above an unknown threshold in a continuous way. The estimators for the threshold and the regression coefficients are obtained using a grid search approach. The asymptotic properties for all the estimators are derived, and the estimator for the threshold is shown to achieve root-n consistency. A weighted CUSUM type test statistic is proposed for the existence of a threshold at a given expectile, and its asymptotic properties are derived under both the null and the local alternative models. This test only requires fitting the model under the null hypothesis in the absence of a threshold, thus it is computationally more efficient than the likelihood-ratio type tests. Simulation studies show that the proposed estimators and test have desirable finite sample performance in both homoscedastic and heteroscedastic cases. The application of the proposed method on a Dutch growth data and a baseball pitcher salary data reveals interesting insights. The proposed method is implemented in the R package cthreshER.

中文翻译：

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连续阈值期望模型

期望回归是探索响应与超出条件均值的解释变量之间关系的有用工具。为建模数据开发了连续阈值期望回归，其中协变量对响应变量的影响是线性的，但在未知阈值以下和以上以连续方式变化。阈值和回归系数的估计是使用网格搜索方法获得的。推导出所有估计量的渐近性质，并显示阈值的估计量以实现根 n 一致性。针对在给定期望值处存在阈值，提出了加权 CUSUM 类型检验统计量，并在零模型和局部替代模型下推导出其渐近特性。这个测试只需要在没有阈值的情况下在零假设下拟合模型，因此它在计算上比似然比类型测试更有效。模拟研究表明，所提出的估计量和检验在同方差和异方差情况下都具有理想的有限样本性能。所提出的方法在荷兰增长数据和棒球投手工资数据上的应用揭示了有趣的见解。所提出的方法在 R 包 cthreshER 中实现。所提出的方法在荷兰增长数据和棒球投手工资数据上的应用揭示了有趣的见解。所提出的方法在 R 包 cthreshER 中实现。所提出的方法在荷兰增长数据和棒球投手工资数据上的应用揭示了有趣的见解。所提出的方法在 R 包 cthreshER 中实现。