Methods for Density Estimation in Thick-Slice Versions of Wicksell's Problem

Abstract

A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the context of the target distribution, rather than its image after the transformation. We apply the new methods, and competing techniques, to a thick-section Wicksell-type problem, using data on the radii of nerve terminals from the electric organ of the electric ray Torpedo marmorata. It is shown that statistical properties of estimators in this problem are very different from those for the thin-slice, classical Wicksell problem, and so the two cases cannot be developed simply by analogy with one another.

Key Words:

• Bandwidth
• Binwidth
• Frequency polygon
• Histogram
• Ill-posed problem
• Inverse problem
• Kernel methods
• Radius
• Sphere
• Stereology
• Thick slice
• Thin slice
• Volterra equation