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性检验是一个经典的统计学问题，具有重要的实际应用，但其基本理论极限尚不明确。本文采用极小极大视角，在均值函数为$\alpha$-H\"{o}lder连续、方差函数为$\beta$-H\"{o}lder连续的框架下考虑该检验问题。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，本文发展了一种基于一阶差分平方的核统计量。值得注意的是，该统计量估计的是一个代理量，而非先前工作中提出的自然二次泛函（即方差函数与其最佳常数$L^2$逼近之间的平方距离）。本文也研究了方差函数无光滑性假设的情形；此时设计点上的方差剖面可以是任意的。尽管缺乏结构，利用噪声的高斯特性，一致性检验仍然是可能的，并且证明其极小极大速率为$n^{-4\alpha} + n^{-1/2}$。利用噪声信息被证明是根本性的必要条件，因为如果对噪声分布仅知其均值为零、方差为一，则一致性检验是不可能的。此外，在方差函数为$\beta$-H\"{o}lder连续但异方差性仅针对设计点进行度量的设定下，当噪声为高斯分布时，极小极大分离速率被证明为$n^{-4\alpha} + n^{-\left((1/2) \vee (4\beta/(4\beta+1))\right)}$；而当噪声分布未知时，该速率为$n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$。