Estimating the Weight Undersize Distribution for the Wicksell problem

In Wicksell's corpuscle problem one is interested in estimating the distribution of sphere diameters from the diameters of circle profiles obtained by a random section of the body containing the sphericle particles. The problem is known to be an ill posed inverse problem. Several regularization techniques have been applied to find solutions. We will review some of these in this article. In practical situations one often is more interested in the distribution of weight rather than the distribution of diameters. In estimating the weight undersite distribution similar problems are encountered. We will consider an estimator that is obtained by smoothing the distribution function of the circle diameters locally. It will be shown how the bandwidth must be chosen to obtain consistency and mean square error optirnality. Also asymptotic normality will be shown.