The Annals of Statistics

Isotonic Estimation and Rates of Convergence in Wicksell's Problem

Piet Groeneboom and Geurt Jongbloed

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Abstract

It is shown that, in the nonparametric setting for the so-called Wicksell problem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than $n^{-1/2}\sqrt{\log n}$. We present an isotonic estimator of the distribution function which attains this rate and derive its asymptotic (normal) distribution. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator.

Article information

Source
Ann. Statist., Volume 23, Number 5 (1995), 1518-1542.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176324310

Digital Object Identifier
doi:10.1214/aos/1176324310

Mathematical Reviews number (MathSciNet)
MR1370294

Zentralblatt MATH identifier
0843.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory

Keywords
Inverse problems minimax rate asymptotic distribution $\arg \max$ functionals

Citation

Groeneboom, Piet; Jongbloed, Geurt. Isotonic Estimation and Rates of Convergence in Wicksell's Problem. Ann. Statist. 23 (1995), no. 5, 1518--1542. doi:10.1214/aos/1176324310. https://projecteuclid.org/euclid.aos/1176324310


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