The Annals of Statistics

Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data

Douglas W. Nychka and Dennis D. Cox

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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.

Article information

Ann. Statist., Volume 17, Number 2 (1989), 556-572.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62J05: Linear regression 41A35: Approximation by operators (in particular, by integral operators) 41A25: Rate of convergence, degree of approximation 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 45L10 45M05: Asymptotics

Method of regularization approximate solution of integral equations rates of convergence nonparametric regression Abel's equation particle size distribution


Nychka, Douglas W.; Cox, Dennis D. Convergence Rates for Regularized Solutions of Integral Equations from Discrete Noisy Data. Ann. Statist. 17 (1989), no. 2, 556--572. doi:10.1214/aos/1176347125.

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