Isotonic Estimation and Rates of Convergence in Wicksell's Problem

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.