Open Access
December 2020 Optimal estimation of variance in nonparametric regression with random design
Yandi Shen, Chao Gao, Daniela Witten, Fang Han
Ann. Statist. 48(6): 3589-3618 (December 2020). DOI: 10.1214/20-AOS1944


Consider the heteroscedastic nonparametric regression model with random design \begin{equation*}Y_{i}=f(X_{i})+V^{1/2}(X_{i})\varepsilon _{i},\quad i=1,2,\ldots ,n,\end{equation*} with $f(\cdot )$ and $V(\cdot )$ $\alpha $- and $\beta $-Hölder smooth, respectively. We show that the minimax rate of estimating $V(\cdot )$ under both local and global squared risks is of the order \begin{equation*}n^{-\frac{8\alpha \beta }{4\alpha \beta +2\alpha +\beta }}\vee n^{-\frac{2\beta }{2\beta +1}},\end{equation*} 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 (Ann. Statist. 36 (2008) 646–664) in a nontrivial 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 $X_{i}$.


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Yandi Shen. Chao Gao. Daniela Witten. Fang Han. "Optimal estimation of variance in nonparametric regression with random design." Ann. Statist. 48 (6) 3589 - 3618, December 2020.


Received: 1 March 2019; Revised: 1 December 2019; Published: December 2020
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185821
Digital Object Identifier: 10.1214/20-AOS1944

Primary: 62G08 , 62G20

Keywords: Minimax rate , Nonparametric regression , random design , U-statistics , variance estimation

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 6 • December 2020
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