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

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, infinity). For that type of deconvolution problems, the usual non-parametric 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. Moreover, this estimator is proved to be strongly consistent.