Open Access
June, 1989 Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data
Douglas W. Nychka, Dennis D. Cox
Ann. Statist. 17(2): 556-572 (June, 1989). DOI: 10.1214/aos/1176347125

Abstract

Given data $y_i = (Kg)(u_i) + \varepsilon_i$ where the $\varepsilon$'s are random errors, the $u$'s are known, $g$ is an unknown function in a reproducing kernel space with kernel $r$ and $K$ is a known integral operator, it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points $\{u_i\}$ become dense in the interval of interest. These rates are shown to depend on the eigenvalue asymptotics of $KRK^\ast$, where $R$ is the integral operator with kernel $r$. The theory is applied to Abel's equation and the estimation of particle size densities in stereology. Rates of convergence of regularized histogram estimates of the particle size density are given.

Citation

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Douglas W. Nychka. Dennis D. Cox. "Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data." Ann. Statist. 17 (2) 556 - 572, June, 1989. https://doi.org/10.1214/aos/1176347125

Information

Published: June, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0672.62054
MathSciNet: MR994250
Digital Object Identifier: 10.1214/aos/1176347125

Subjects:
Primary: 62G05
Secondary: 41A25 , 41A35 , 45L10 , 45M05 , 47A53 , 62J05

Keywords: Abel's equation , approximate solution of integral equations , method of regularization , Nonparametric regression , particle size distribution , rates of convergence

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 2 • June, 1989
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