Abstract
Consider a $p$-times differentiable unknown regression function $\\theta$ of a $d$-dimensional measurement variable. Let $T(\\theta)$ denote a derivative of $\\theta$ of order $m$ and set $r = (p - m)/(2p + d)$. Let $\\hat{T}_n$ denote an estimator of $T(\\theta)$ based on a training sample of size $n$, and let $\\| \\hat{T}_n - T(\\theta)\\|_q$ be the usual $L^q$ norm of the restriction of $\\hat{T}_n - T(\\theta)$ to a fixed compact set. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for $\\| \\hat{T}_n - T(\\theta)\\|_q$ is $n^{-r}$ if $0 < q < \\infty$; while $(n^{-1} \\log n)^r$ is the optimal rate if $q = \\infty$.
Keywords
MathematicsDifferentiable functionCombinatoricsEstimatorRate of convergenceNonparametric regressionOrder (exchange)Regression functionMathematical analysisStatistics
Related Publications
A Strong Law for the Longest Edge of the Minimal Spanning Tree
Suppose $X_1, X_2, X_3,\\ldots$ are independent random points in $\\mathbf{R}^d,d\\geq 2$, with common density $f$, having connected compact support $\\Omega$ with smooth bounda...
Maximum Likelihood Estimation in Truncated Samples
In this paper we consider the problem of estimation of parameters from a sample in which only the first $r$ (of $n$) ordered observations are known. If $r = \\lbrack qn \\rbrack...
Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results
Let ${M}$ be a compact Riemannian submanifold of ${{\\bf R}^m}$ of dimension\n$\\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d. points in ${M}$\nwith uniform dist...
Multivariate Smoothing Spline Functions
Given data $z_i = g(t_i ) + \varepsilon _i , 1 \leqq i \leqq n$, where g is the unknown function, the $t_i $ are known d-dimensional variables in a domain $\Omega $, and the $\v...
The diffraction of X-rays by distorted-crystal aggregates. II. Diffraction by bent lamellae
The apparent particle size of distorted crystallites varies from independence of θ and λ (particle-size broadening) to proportionality with λ cosec θ (distortion broadening) as ...
Publication Info
- Year
- 1982
- Type
- article
- Volume
- 10
- Issue
- 4
- Citations
- 1477
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
1477
OpenAlex
Cite This
Charles J. Stone
(1982).
Optimal Global Rates of Convergence for Nonparametric Regression.
The Annals of Statistics
, 10
(4)
.
https://doi.org/10.1214/aos/1176345969
Identifiers
- DOI
- 10.1214/aos/1176345969