Wicksell's Problem Research Articles

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/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.

Estimation of the Size Distribution of Closed Cell Elements from Analysis of Their Random Plane Sections

Numerical estimates of the size distribution of spherical particles derived from the distribution of their random plane section sizes (Wicksell's problem) are well known. The present paper considers problems in which the observed particles are approximated by either three-axial or ''crushed'' ellipsoids. Based on the described numerical solutions of the problems, the size distributions of some important neuron ultrastructural elements are estimated from an electron microscopic morphometrical study. Experimental evidence supporting the ''crushed'' ellipsoid paradigm is discussed.

Theoretical aspects of ill-posed problems in statistics

Ill-posed problems arise in a wide variety of practical statistical situations, ranging from biased sampling and Wicksell's problem in stereology to regression, errors-in-variables and empirical Bayes models. The common mathematics behind many of these problems is operator inversion. When this inverse is not continuous a regularization of the inverse is needed to construct approximate solutions. In the statistical literature, however, ill-posed problems are rather often solved in an ad hoc manner which obccures these common features. It is our purpose to place the concept of regularization within a general and unifying framework and to illustrate its power in a number of interesting statistical examples. We will focus on regularization in Hilbert spaces, using spectral theory and reduction to multiplication operators. A partial extension to a Banach function space is briefly considered.

Wicksell's corpuscle problem in spherical spaces

Wicksell's problem of unfolding ball size distributions is solved for the case of spherical spaces of curvature K > 0. The well-known formulas for the euclidean case result as limits if K → 0. The paper is completed by a brief discussion of some computer-aided statistical procedures.

Wicksell's corpuscle problem in spherical spaces

Wicksell's problem of unfolding ball size distributions is solved for the case of spherical spaces of curvature K > 0. The well-known formulas for the euclidean case result as limits if K → 0. The paper is completed by a brief discussion of some computer-aided statistical procedures.