A Kiefer – Wolfowitz Comparison Theorem for Wicksell ’ S Problem

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. 1. Introduction. Our starting point is Groeneboom and Jongbloed's [5] analysis of Wicksell's [12] " Corpuscle Problem, " in anatomy: Given cross-sections of a large number of corpuscles of different sizes, the distribution of radii of corpuscles was to be estimated. Assuming corpuscles were spherical, the relation between the distribution of the corpuscle radii and the distribution of the observable circular sections was derived. Wicksell approximated the density of spherical radii by a step function and then used the distri-butional relationship between spherical radii and circular radii to estimate the distribution of spherical radii. Groeneboom and Jongbloed [5] showed how isotonic techniques can be used in Wicksell's problem. They related the distribution of spherical radii to a nonincreasing function that can be estimated unbiasedly. The unbiased estimate was not monotone, however, and they showed how it can be improved by imposing the shape restriction. Here we extend the isotonic analysis to estimate a regression function. The extension is motivated by the problem of estimating the velocity dispersions in astronomy which, in turn, is motivated by the problem of estimating the distribution of dark matter, as explained in [11].) denote the three-dimensional position and velocity of