Isotonic estimation and rates of convergence in Wicksell's problem (Q1914256)

\textit{D. S. Wicksel} [Biometrika 17, 84-99 (1925)] studied the following stereological problem: Suppose that a number of spheres are embedded in an opaque medium. The item of interest is the distribution function of the sphere radii. Since the medium is opaque, we cannot observe a sample of sphere radii directly. What we can observe is a cross section of the medium, showing circular sections of some spheres. Here it is shown that, in the nonparametric setting for the 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.