MATHEMATICA BOHEMICA, Vol. 128, No. 4, pp. 419-438 (2003)

Stereology of extremes; size of spheroids

Daniel Hlubinka

Daniel Hlubinka, Univerzita Karlova v Praze, Matematicko-fyzikalni fakulta, katedra pravdepodobnosti a statistiky, Sokolovska 83, 186 75 Praha 8, e-mail:

Abstract: The prediction of size extremes in Wicksell's corpuscle problem with oblate spheroids is considered. Three-dimensional particles are represented by their planar sections (profiles) and the problem is to predict their extremal size under the assumption of a constant shape factor. The stability of the domain of attraction of the size extremes is proved under the tail equivalence condition. A simple procedure is proposed of evaluating the normalizing constants from the tail behaviour of appropriate distribution functions and its results are employed for the estimation of the spheroid size. Examples covering families of Gamma, Pareto and Weibull distributions are provided. A short discussion of maximum likelihood estimators of the normalizing constants is also included.

Keywords: sample extremes, domain of attraction, normalizing constants, FGM system of distributions

Classification (MSC2000): 60G70, 62G32, 62P30

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