Isotonic estimation and rates of convergence in Wicksell's problem
type: Item
instance of: scholarly article
title: Isotonic estimation and rates of convergence in Wicksell's problem
zbMATH Open document ID: 0843.62034
author: Piet Groeneboom
author: Geurt Jongbloed
published in: The Annals of Statistics
publication date: 1996-08-22T00:00:00Z
review text: \textit{D. S. Wicksel} [Biometrika 17, 84-99 (1925)] studied the following stereological problem: Suppose that a number of spheres are embedded in an opaque medium. The item of interest is the distribution function of the sphere radii. Since the medium is opaque, we cannot observe a sample of sphere radii directly. What we can observe is a cross section of the medium, showing circular sections of some spheres. Here it is shown that, in the nonparametric setting for the 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.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 885106
zbMATH Keywords: inverse problem
zbMATH Keywords: minimax rate
zbMATH Keywords: arg max functionals
zbMATH Keywords: asymptotic normality
zbMATH Keywords: isotonic estimator
zbMATH Keywords: variance
zbMATH Keywords: limiting distribution
zbMATH Keywords: asymptotic variance
zbMATH Keywords: plug-in estimator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176324310
OpenAlex ID: W2022931946
DOI: 10.1214/AOS/1176324310
description: scientific article; zbMATH DE number 885106