A Kiefer--Wolfowitz comparison theorem for Wicksell's problem

Details A Kiefer--Wolfowitz comparison theorem for Wicksell's problem A Kiefer--Wolfowitz comparison theorem for Wicksell's problem

2006-11-03

arxiv.org/abs/math/0611088v2

Annals of Statistics 2007, Vol. 35, No. 4, 1559-1575

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/009053606000001604 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000001604

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.

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