Optimal estimation of variance in nonparametric regression with random design
Optimal estimation of variance in nonparametric regression with random design
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DOI：
10.1214/20-aos1944
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发表时间：
2019-02
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通讯作者：
Yandi Shen;Chao Gao;D. Witten;Fang Han
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作者：
Yandi Shen;Chao Gao;D. Witten;Fang Han
考虑随机设计的异方差非参数回归模型\begin{align*} Y_i = f(X_i) + V^{1/2}(X_i)\varepsilon_i, \quad i=1,2,\ldots,n, \end{align*}，分别为$f(\cdot)$和$V(\cdot)$$\alpha$ -和$\beta$ -Hölder光滑。我们证明了在局部和全局风险平方下估计$V(\cdot)$的最小最大率为\begin{align*} n^{-\frac{8\alpha\beta}{4\alpha\beta + 2\alpha + \beta}} \vee n^{-\frac{2\beta}{2\beta+1}}, \end{align*}，其中$a\vee b := \max\{a,b\}$对于任意两个实数$a,b$。这一结果扩展了Wang等人[2008]推导出的固定设计率$n^{-4\alpha} \vee n^{-2\beta/(2\beta+1)}$，从第一项中$\alpha$和$\beta$的出现可以看出。在方差不变的特殊情况下，我们证明方差估计的极大极小率为$n^{-8\alpha/(4\alpha+1)}\vee n^{-1}$，这进一步表明二次泛函估计的极大极小率相同，从而将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大极小率统一起来。为了实现极大极小率，我们开发了一个基于u统计量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造的。
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 $X_i$.