This paper studies minimax rates of convergence for nonparametric location-scale models, which include mean, quantile and expectile regression settings. Under Hellinger differentiability on the error distribution and other mild conditions, we show that the minimax rate of convergence for estimating the regression function under the squared $L_2$ loss is determined by the metric entropy of the nonparametric function class. Different error distributions, including asymmetric Laplace distribution, asymmetric connected double truncated gamma distribution, connected normal-Laplace distribution, Cauchy distribution and asymmetric normal distribution are studied as examples. Applications on low order interaction models and multiple index models are also given.

翻译：本文研究了非参数位置尺度模型的极小极大收敛速度，该类模型涵盖均值回归、分位数回归及期望回归设定。在误差分布满足Hellinger可微性及其他温和条件下，我们证明：在平方$L_2$损失下，回归函数估计的极小极大收敛速度由非参数函数类的度量熵决定。本文以不对称拉普拉斯分布、不对称连通双截断伽马分布、连通正态-拉普拉斯分布、柯西分布及不对称正态分布等不同误差分布为例展开研究，并给出了在低阶交互模型与多指标模型中的应用。