The Annals of Statistics

A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem

Xiao Wang and Michael Woodroofe

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We extend the isotonic analysis for Wicksell’s problem to estimate a regression function, which is motivated by the problem of estimating dark matter distribution in astronomy. The main result is a version of the Kiefer–Wolfowitz theorem comparing the empirical distribution to its least concave majorant, but with a convergence rate $n^{-1}\log n$ faster than $n^{-2/3}\log n$. The main result is useful in obtaining asymptotic distributions for estimators, such as isotonic and smooth estimators.

Article information

Ann. Statist., Volume 35, Number 4 (2007), 1559-1575.

First available in Project Euclid: 29 August 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G08: Nonparametric regression 62G20: Asymptotic properties

dark matter empirical processes isotonic estimation least concave majorant regression function velocity dispersions


Wang, Xiao; Woodroofe, Michael. A Kiefer–Wolfowitz comparison theorem for Wicksell’s problem. Ann. Statist. 35 (2007), no. 4, 1559--1575. doi:10.1214/009053606000001604.

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