Adaptive and Efficient Isotonic Estimation in Wicksell's Problem

Abstract

We consider nonparametric estimation in Wicksell’s problem, which has applications in astronomy for estimating the distribution of star positions in a galaxy and in material sciences for determining a material’s 3D microstructure from 2D cross sections. We focus on the isotonised version of the plug-in estimator (IIE) for the cdf F of the spheres’ squared radii. This estimator is fully automatic, requiring no tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function F to be estimated. We also prove a local asymptotic minimax lower bound in this non-standard setting, with √log⁡𝑛/𝑛-asymptotics and where the functional F to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator.