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 $X_i$.

中文翻译：

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随机设计非参数回归中方差的最优估计

我们表明方差估计的极小极大率是 $n^{-8\alpha/(4\alpha+1)}\vee n^{-1}$，这进一步暗示了二次函数估计的相同率，因此统一非参数回归模型下的极大极小率与密度模型和白噪声模型下的极小极大率。为了实现最小最大速率，我们开发了一个基于 U 统计的局部多项式估计器和一个下界，该下界是在为 $\varepsilon_i$ 和 $X_i$ 设计的指定随机分布族上构建的。