Statistica Sinica 28 (2018), 93-113

NONPARAMETRIC CONFIDENCE BANDS

IN WICKSELL'S PROBLEM

Jakub Wojdyła and Zbigniew Szkutnik

AGH University of Science and Technology

Abstract: Although asymptotic nonparametric confidence bands have been constructed in the last decade in some inverse problems, like density deconvolution, inverse regression with a convolution operator, and regression with errors in variables, there seems to be no such construction for practically important inverse problems of stereology. Working with a kernel-type nonparametric estimator of the density of squared radii in the stereological Wicksell's problem, we partially fill this gap by constructing some corresponding asymptotic uniform confidence bands and an automatic bandwidth selection method, tuned to perform well in finite samples in terms of both area and coverage probability of the confidence bands. The performance of the new procedures is investigated in simulations and demonstrated with some astronomical data related to the M62 globular cluster.

Key words and phrases: Abel integral equation, ill-posed inverse problem, kernel methods, nonparametric curve estimation, stereology.