  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.
