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Statistica Sinica 18(2008), 1063-1080





MINIMAX UNFOLDING OF THE SPHERES

SIZE DISTRIBUTION FROM LINEAR SECTIONS


Anna Dudek and Zbigniew Szkutnik


AGH University of Science and Technology


Abstract: The stereological problem of unfolding the spheres size distribution from linear sections is analysed as a statistical inverse problem of estimation of a Poisson process intensity function from indirectly observed and binned data. Using suitably constructed singular value decomposition of the folding operator, a spectral estimator is constructed that is, up to a logarithmic factor, asymptotically rate minimax over a Sobolev-type class of functions. Finite sample behaviour of the estimator is demonstrated in a small numerical experiment.



Key words and phrases: Discretization, empirical risk minimization, ill-posed inverse problem, rate minimaxity, singular value decomposition, stereology.

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