Asymptotic confidence bands in the Spektor-Lord-Willis problem via kernel estimation of intensity derivative

Abstract

The stereological problem of unfolding the distribution of spheres radii from linear sections, known as the Spektor-Lord-Willis problem, is formulated as a Poisson inverse problem and an $L^{2}$-rate-minimax solution is constructed over some restricted Sobolev classes. The solution is a specialized kernel-type estimator with boundary correction. For the first time for this problem, non-parametric, asymptotic confidence bands for the unfolded function are constructed. Automatic bandwidth selection procedures based on empirical risk minimization are proposed. It is shown that a version of the Goldenshluger-Lepski procedure of bandwidth selection ensures adaptivity of the estimators to the unknown smoothness. The performance of the procedures is demonstrated in a Monte Carlo experiment.