## The Annals of Statistics

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

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

Douglas W. Nychka and Dennis D. Cox

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347125

**Digital Object Identifier**

doi:10.1214/aos/1176347125

**Mathematical Reviews number (MathSciNet)**

MR994250

**Zentralblatt MATH identifier**

0672.62054

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176347125