Expectile regression, in contrast to classical linear regression, allows for heteroscedasticity and omits a parametric specification of the underlying distribution. This model class can be seen as a quantile-like generalization of least squares regression. Similarly as in quantile regression, the whole distribution can be modeled with expectiles, while still offering the same flexibility in the use of semiparametric predictors as modern mean regression. However, even with no parametric assumption for the distribution of the response in expectile regression, the model is still constructed with a linear relationship between the fitted value and the predictor. If the true underlying relationship is nonlinear then severe biases can be observed in the parameter estimates as well as in quantities derived from them such as model predictions. We observed this problem during the analysis of the distribution of a self-reported hearing score with limited range. Classical expectile regression should in theory adhere to these constraints, however, we observed predictions that exceeded the maximum score. We propose to include a response function between the fitted value and the predictor similarly as in generalized linear models. However, including a fixed response function would imply an assumption on the shape of the underlying distribution function. Such assumptions would be counterintuitive in expectile regression. Therefore, we propose to estimate the response function jointly with the covariate effects. We design the response function as a monotonically increasing P-spline, which may also contain constraints on the target set. This results in valid estimates for a self-reported listening effort score through nonlinear estimates of the response function. We observed strong associations with the speech reception threshold.

中文翻译：

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具有灵活响应函数的广义期望回归

与经典线性回归相比，期望回归允许异方差性并省略潜在分布的参数规范。这个模型类可以看作是最小二乘回归的分位数泛化。与分位数回归类似，整个分布可以用期望值建模，同时在使用半参数预测变量方面仍提供与现代平均回归相同的灵活性。然而，即使在期望回归中没有对响应分布的参数假设，该模型仍然是用拟合值和预测变量之间的线性关系构建的。如果真正的潜在关系是非线性的，那么可以在参数估计以及从它们导出的数量（例如模型预测）中观察到严重的偏差。我们在分析范围有限的自我报告听力分数的分布时发现了这个问题。理论上，经典的期望回归应该遵守这些约束，但是，我们观察到超过最大分数的预测。我们建议在拟合值和预测变量之间包含一个响应函数，类似于广义线性模型。但是，包括固定响应函数将意味着对基础分布函数形状的假设。这种假设在预期回归中是违反直觉的。因此，我们建议结合协变量效应来估计响应函数。我们将响应函数设计为单调递增的 P 样条，它也可能包含对目标集的约束。这导致通过响应函数的非线性估计对自我报告的听力努力分数进行有效估计。我们观察到与语音接收阈值的强关联。