Estimation of the density of regression errors
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回归误差密度的估计
- 研究问题
- 在非参数回归中误差无法直接观测时，能否用估计残差构造误差密度估计，并达到知道真实误差的预言机精度？
- 方法与思路
- 作者以平均积分平方误差为准则，将代入残差用于分块收缩的Efromovich–Pinsker密度估计，并讨论异方差及依赖情形的扩展。
- 材料陈述的贡献
- 摘要称在设计密度与回归函数满足一定可微条件下，估计达到直接观测误差时的极小极大速率及精确常数，适用于有限和无限支持。
- 阅读关注点
- 应核对可微性要求、回归估计误差的控制及误差分布类别，并区分基础模型中的误差密度与依赖扩展中估计的边际误差密度。
- 阅读边界
- 仅据版本未注明的OpenAlex索引摘要，具体光滑条件和扩展边界未给出；不能据此认为任意残差密度估计均有预言机性质。
原文摘要
Estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach. This article uses a nonparametric approach with the mean integrated squared error (MISE) criterion. It solves a long-standing problem, formulated two decades ago by Mark Pinsker, about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying regression errors. The solution implies that, under a mild assumption on the differentiability of the design density and regression function, the MISE of a data-driven error density estimator attains minimax rates and sharp constants known for the case of directly observed regression errors. The result holds for error densities with finite and infinite supports. Some extensions of this result for more general heteroscedastic models with possibly dependent errors and predictors are also obtained; in the latter case the marginal error density is estimated. In all considered cases a blockwise-shrinking Efromovich–Pinsker density estimate, based on plugged-in residuals, is used. The obtained results imply a theoretical justification of a customary practice in applied regression analysis to consider residuals as proxies for underlying regression errors. Numerical and real examples are presented and discussed, and the S-PLUS software is available.