Nonparametric confidence bands in Wicksell's problem

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.

A note on confidence intervals for deblurred images

“…5.1.3] and [6]. The latter paper was also the first step towards the construction of confidence bands in inverse problems and was followed in recent years by several similar works [2,4,5,10,11,15,18,19].…”