Abstract
In the stereological unfolding problem of size and shape factor of spheroidal particles the extremal shape factor is investigated. The transformation of particle parameters is shown to be stable with respect to the domain of attraction in three situations: (a) Conditional distribution of shape factor given particle size. (b) Conditional distribution of shape factor given particle section size. (c) Marginal distribution of shape factor. In situation (a) the normalizing constants are derived in a parametric model, which enables prediction of extreme shape factor after a preliminary solution of the size unfolding problem. In both situations (b) and (c) obtaining a similar procedure is still an open problem. The uniformity conditions necessary for (b) and (c) are shown to be mild.
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References
Beneš, V., Jiruše, M., and Slámová, M., “Stereological unfolding of the trivariate size-shape-orientation distribution of spheroidal particles with application,” Acta Materialia 45(3), 1105–1197, (1997).
Cruz-Orive, L.-M., “Particle size-shape distributions; The general spheroid problem,” J. Microscopy 107(3), 235–253, (1976).
Drees, H. and Reiss, R.-D., “Tail behavior in Wicksell's Corpuscle problem,” Probability Theory and Applications (J. Galambos, J. Kátai, eds.), Kluwer, Dordrecht, (1992).
de Haan, L., On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Math. Centre Tracts 32, Mathematisch Centrum, Amsterdam, (1975).
Ohser, J. and Mücklich, F., Statistical Analysis of Microstructures in Material Science, Wiley, New York, 2000.
Stoyan, D., Kendall, W.S., and Mecke, J., Stochastic Geometry And Its Applications, (second edition), Wiley, New York, 1995.
Takahashi, R., “Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution,” Stat. Prob. Letters 5, 197–200, (1987).
Takahashi, R. and Sibuya, M., “The maximum size of the planar sections of random spheres and its application to metallurgy,” Ann. Inst. Statist. Math. 48(1), 361–377, (1996).
Takahashi, R. and Sibuya, M., “Prediction of the maximum size in Wicksell's Corpuscle problem,” Ann. Inst. Statist. Math. 50(2), 361–377, (1998).
Weissman, I., “Estimation of parameters and large quantiles based on the k largest observations,” J. American Stat. Assoc. 73(364), 812–815.
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Hlubinka, D. Stereology of Extremes; Shape Factor of Spheroids. Extremes 6, 5–24 (2003). https://doi.org/10.1023/A:1026234329084
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DOI: https://doi.org/10.1023/A:1026234329084