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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.

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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

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 2 • June, 1989
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