Opuscula Math. 27, no. 1 (2007), 151-165

Opuscula Mathematica

Unfolding spheres size distribution from linear sections with B-splines and EMDS algorithm

Zbigniew Szkutnik

Abstract. The stereological problem of unfolding spheres size distribution from linear sections is formulated as a problem of inverse estimation of a Poisson process intensity function. A singular value expansion of the corresponding integral operator is given. The theory of recently proposed $$B$$-spline sieved quasi-maximum likelihood estimators is modified to make it applicable to the current problem. Strong $$L^2$$-consistency is proved and convergence rates are given. The estimators are implemented with the recently proposed EMDS algorithm. Promising performance of this new methodology in finite samples is illustrated with a numerical example. Data grouping effects are also discussed.

Keywords: inverse problem, singular value expansion, stereology, discretization, quasi-maximum likelihood estimator.

Mathematics Subject Classification: 62G05, 45Q05.

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