Sieved maximum likelihood estimation in Wicksell's problem and related deconvolution problems (Q2722308)

The problem that is known under the name ``Wickell's corpuscle problem'' has a history of more than 70 years. Its name comes from the Swedish mathematician Sven Wicksell from Lund University, who constructed a mathematical model for an anatomical problem [Biometrika 17, 84-99 (1925)]. The anatomical problem is the following: In order to get information concerning the sizes of (approximately spherical) ``follicles'' in human spleens, post-mortem examinations are performed. The spleens are cut in slices (at relatively large mutual distance) and a number of (approximately circular) profiles of these follicles are observed. In this paper it is shown that the classical Wicksell problem is related to a deconvolution problem where the convolution kernel is unbounded, convex and decreasing on \((0,\infty).\) For that type of deconvolution problems the usual nonparametric maximum likelihood estimator of the distribution function is shown not to exist. A sieved maximum likelihood estimator is defined and some algorithms are described that can be used to compute this estimator. This estimator is proved to be strongly consistent.