  Consider the heteroscedastic nonparametric regression model with random
design \begin{align*} Y_i = f(X_i) + V^{1/2}(X_i)\varepsilon_i, \quad
i=1,2,\ldots,n, \end{align*} with $f(\cdot)$ and $V(\cdot)$ $\alpha$- and
$\beta$-H\"older smooth, respectively. We show that the minimax rate of
estimating $V(\cdot)$ under both local and global squared risks is of the order
\begin{align*} n^{-\frac{8\alpha\beta}{4\alpha\beta + 2\alpha + \beta}} \vee
n^{-\frac{2\beta}{2\beta+1}}, \end{align*} where $a\vee b := \max\{a,b\}$ for
any two real numbers $a,b$. This result extends the fixed design rate
$n^{-4\alpha} \vee n^{-2\beta/(2\beta+1)}$ derived in Wang et al. [2008] in a
non-trivial manner, as indicated by the appearances of both $\alpha$ and
$\beta$ in the first term. In the special case of constant variance, we show
that the minimax rate is $n^{-8\alpha/(4\alpha+1)}\vee n^{-1}$ for variance
estimation, which further implies the same rate for quadratic functional
estimation and thus unifies the minimax rate under the nonparametric regression
model with those under the density model and the white noise model. To achieve
the minimax rate, we develop a U-statistic-based local polynomial estimator and
a lower bound that is constructed over a specified distribution family of
randomness designed for both $\varepsilon_i$ and $X_i$.
