Revisiting the Wicksel problem

Aad van der Vaart

TU Delft

 

We consider estimating the distribution of radii of balls distributed randomly in a three-dimensional (opaque) medium from the observed radii of circles in a two-dimensional intersection. Wicksel's inversion formula (1925) relates the distributions of the two radii, correcting both for the fact that the observed circle radii are smaller than the radii of the corresponding balls and the fact that balls of a bigger radii are intersected more often and hence are over represented in the data. We briefly discuss some extensions and applications. Next for the classical Wicksel problem, we show new results on the so-called isotonic estimator, showing that this estimator is adaptive to the smoothness of the true distriibution and asymptotically efficient in the semiparametric sense. We also consider two Bayesian approaches both employing a Dirichlet prior on an unknown distribution. For one approach we show a Bernstein-von Mises theorem (at the unusual square of n/(log n) rate). 

For the other approach, which is fully Bayesian, we show by picture that it seems to work as well, but present only posterior consistency results. 

[Based on joint work with Francesco Gili and Geurt Jongbloed.]