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Title: Asymptotic confidence bands in the Spektor-Lord-Willis problem Authors:  Bogdan Cmiel - Polish Academy of Sciences (Poland)
Zbigniew Szkutnik - AGH University of Science and Technology (Poland) [presenting]
Jakub Wojdyla - AGH University of Science and Technology (Poland)
Abstract: Confidence bands for a density function of directly observed i.i.d. data have been proposed since 1973. In the last decade, asymptotic nonparametric confidence bands have been constructed in some inverse problems, like density deconvolution, inverse regression with a convolution operator and regression with errors in variables. There seems to have been, however, no such construction for practically important inverse problems of stereology. This gap will be partially filled by constructing a kernel-type estimator for the density of balls radii in the stereological Spektor-Lord-Willis (SLW) problem, along with corresponding asymptotic uniform confidence bands and an automatic bandwidth selection method. Recall that the SLW problem consists in unfolding the distribution of random radii of balls randomly placed in an opaque medium and only observed as line segments on a random line section through the medium (like drilling through a rock or a muscle biopsy). The problem will be formulated as an ill-posed Poisson inverse problem and the construction of asymptotic confidence bands will be based, as in earlier contributions, on the supremum of a stationary Gaussian process. The finite-sample performance of the new procedures will be demonstrated in a simulation experiment.