Title: Estimating a regression function under possible heteroscedastic and heavy-tailed errors. Application to shape-restricted regression Show Chinese title

Title: 在可能存在异方差和重尾误差的情况下估计回归函数。 应用于受限形状的回归。

Abstract: We consider a regression framework where the design points are deterministic and the errors possibly non-i.i.d. and heavy-tailed (with a moment of order $p$ in $[1,2]$). Given a class of candidate regression functions, we propose a surrogate for the classical least squares estimator (LSE). For this new estimator, we establish a nonasymptotic risk bound with respect to the absolute loss which takes the form of an oracle type inequality. This inequality shows that our estimator possesses natural adaptation properties with respect to some elements of the class. When this class consists of monotone functions or convex functions on an interval, these adaptation properties are similar to those established in the literature for the LSE. However, unlike the LSE, we prove that our estimator remains stable with respect to a possible heteroscedasticity of the errors and may even converge at a parametric rate (up to a logarithmic factor) when the LSE is not even consistent. We illustrate the performance of this new estimator over classes of regression functions that satisfy a shape constraint: piecewise monotone, piecewise convex/concave, among other examples. The paper also contains some approximation results by splines with degrees in $\{0,1\}$ and VC bounds for the dimensions of classes of level sets. These results may be of independent interest. Show Chinese abstract

Abstract: 我们考虑一个回归框架，其中设计点是确定性的，误差可能是非独立同分布的，并且具有重尾性（在 $[1,2]$ 阶矩下有 $p$ 阶矩）。 给定一组候选回归函数类，我们提出了经典最小二乘估计量（LSE）的一个替代估计量。对于这个新估计量，我们建立了关于绝对损失的非渐近风险界，该界以类似于 oracle 不等式的形式呈现。这一不等式表明我们的估计量相对于该类的一些元素具有自然的自适应性质。当此类由单调函数或区间上的凸函数组成时，这些自适应性质与文献中针对 LSE 建立的性质类似。然而，与 LSE 不同的是，我们证明了我们的估计量在误差可能具有异方差性时仍然保持稳定，并且在 LSE 甚至不一致的情况下，我们的估计量可能以接近参数速率收敛（至多带有一个对数因子）。我们通过满足形状约束的回归函数类（例如分段单调、分段凸/凹等例子）展示了此新估计量的表现。 本文还包含了关于 $\{0,1\}$ 次幂样条逼近的结果以及水平集类维数的 VC 边界。这些结果可能具有独立的兴趣价值。