Isotonic Estimation and Rates of Convergence in Wicksell's Problem

摘要

It is shown that, in the nonparametric setting for the so-called Wicksellproblem, the distribution function of the squared radii of the balls cannot be estimated at a rate faster than (1/square root of n)(square root of log n). An isotonic estimator of the distribution function which attains this rate is presented. Its asymptotic distribution is derived. It is shown that the variance of this limiting distribution is exactly half the asymptotic variance of the naive plug-in estimator. The violation of both monotonicity and the range constraints for distribution functions by the estimators are discussed. An isotonized naive estimator is defined.