Heteroskedasticity testing in nonparametric regression is a classic statistical problem with important practical applications, yet fundamental limits are unknown. Adopting a minimax perspective, this article considers the testing problem in the context of an $\alpha$-H\"{o}lder mean and a $\beta$-H\"{o}lder variance function. For $\alpha > 0$ and $\beta \in (0, 1/2)$, the sharp minimax separation rate $n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$ is established. To achieve the minimax separation rate, a kernel-based statistic using first-order squared differences is developed. Notably, the statistic estimates a proxy rather than a natural quadratic functional (the squared distance between the variance function and its best $L^2$ approximation by a constant) suggested in previous work. The setting where no smoothness is assumed on the variance function is also studied; the variance profile across the design points can be arbitrary. Despite the lack of structure, consistent testing turns out to still be possible by using the Gaussian character of the noise, and the minimax rate is shown to be $n^{-4\alpha} + n^{-1/2}$. Exploiting noise information happens to be a fundamental necessity as consistent testing is impossible if nothing more than zero mean and unit variance is known about the noise distribution. Furthermore, in the setting where the variance function is $\beta$-H\"{o}lder but heteroskedasticity is measured only with respect to the design points, the minimax separation rate is shown to be $n^{-4\alpha} + n^{-\left((1/2) \vee (4\beta/(4\beta+1))\right)}$ when the noise is Gaussian and $n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$ when the noise distribution is unknown.

翻译：非参数回归中的异方差检验是一个经典的统计问题，具有重要的实际应用价值，但其基本极限尚不明确。本文从极小极大视角出发，在均值函数为$\alpha$-Hölder光滑、方差函数为$\beta$-Hölder光滑的设定下研究该检验问题。对于$\alpha > 0$和$\beta \in (0, 1/2)$，本文建立了尖锐的极小极大分离速率$n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$。为达到该极小极大分离速率，本文提出了一种基于一阶平方差的核统计量。值得注意的是，该统计量估计的是一个代理量，而非前期工作中建议的自然二次泛函（方差函数与其在$L^2$最佳常数逼近之间的平方距离）。本文还研究了方差函数不假设光滑性的情形，此时设计点上的方差轮廓可以是任意的。尽管缺乏结构，利用噪声的高斯特性仍可实现一致检验，且极小极大速率被证明为$n^{-4\alpha} + n^{-1/2}$。利用噪声信息恰恰是根本性的必要条件：若对噪声分布仅知道零均值和单位方差，则无法实现一致检验。此外，在方差函数为$\beta$-Hölder光滑但异质性仅相对于设计点度量的设定下，当噪声为高斯分布时，极小极大分离速率被证明为$n^{-4\alpha} + n^{-\left((1/2) \vee (4\beta/(4\beta+1))\right)}$；当噪声分布未知时，则为$n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$。