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sample extremes; domain of attraction; normalising constants; FGM system of distributions
The extremal shape factor of spheroidal particles is studied. Three dimensional particles are considered to be observed via their two dimensional profiles and the problem is to predict the extremal shape factor in a given size class. We proof the stability of the domain of attraction of the spheroid’s and its profile shape factor under a tail equivalence condition. We show namely that the Farlie–Gumbel–Morgenstern bivariate distributions gives the tail uniformity. We provide a way how to find normalising constants for the shape factor extremes. The theory is illustrated on examples of distributions belonging to Gumbel and Fréchet domain of attraction. We discuss the ML estimator based on the largest observations and hence the possible statistical applications at the end.
[1] Beneš V., Bodlák, K., Hlubinka D.: Stereology of extremes; FGM bivariate distributions. Method. Comput. Appl. Probab. 5 (2003), 289–308 DOI 10.1023/A:1026283103180 | MR 2016768
[2] Beneš V., Jiruše, M., Slámová M.: Stereological unfolding of the trivariate size-shape-orientation distribution of spheroidal particles with application. Acta Materialia 45 (1997), 1105–1197 DOI 10.1016/S1359-6454(96)00249-2
[3] Cruz-Orive L.-M.: Particle size-shape distributions; the general spheroid problem. J. Microsc. 107 (1976), 235–253 DOI 10.1111/j.1365-2818.1976.tb02446.x
[4] Drees H., Reiss R.-D.: Tail behavior in Wicksell’s corpuscle problem. In: Probability Theory and Applications (J. Galambos and J. Kátai, eds.), Kluwer, Dordrecht 1992, pp. 205–220 MR 1211909
[5] Embrechts P., Klüppelberg, C., Mikosch T.: Modelling Extremal Events for Insurance and Finance. Springer–Verlag, Berlin 1997 MR 1458613 | Zbl 0873.62116
[6] Haan L. de: On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. (Mathematical Centre Tract 32.) Mathematisch Centrum Amsterdam, 1975 MR 0286156 | Zbl 0226.60039
[7] Hlubinka D.: Stereology of extremes; shape factor of spheroids. Extremes 5 (2003), 5–24 DOI 10.1023/A:1026234329084 | MR 2021590 | Zbl 1051.60011
[8] Hlubinka D.: Stereology of extremes; size of spheroids. Mathematica Bohemica 128 (2003), 419–438 MR 2032479 | Zbl 1053.60053
[9] Ohser J., Mücklich F.: Statistical Analysis of Microstructures in Materials Science. Wiley, New York 2000 Zbl 0960.62129
[10] Reiss R.-D.: A Course on Point Processes. Springer–Verlag, Berlin 1993 MR 1199815 | Zbl 0771.60037
[11] Reiss R.-D., Thomas M.: Statistical Analysis of Extreme Values. From Insurance, Finance, Hydrology and Other Fields. Second edition. Birkhäuser, Basel 2001 MR 1819648 | Zbl 1122.62036
[12] Takahashi R.: Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution. Statist. Probab. Lett. 5 (1987), 197–200 DOI 10.1016/0167-7152(87)90039-3 | MR 0881196 | Zbl 0617.62050
[13] Takahashi R., Sibuya M.: The maximum size of the planar sections of random spheres and its application to metalurgy. Ann. Inst. Statist. Math. 48 (1996), 127–144 DOI 10.1007/BF00049294 | MR 1392521
[14] Takahashi R., Sibuya M.: Prediction of the maximum size in Wicksell’s corpuscle problem. Ann. Inst. Statist. Math. 50 (1998), 361–377 DOI 10.1023/A:1003451417655 | MR 1868939 | Zbl 0986.62075
[15] Takahashi R., Sibuya M.: Prediction of the maximum size in Wicksell’s corpuscle problem. II. Ann. Inst. Statist. Math. 53 (2001), 647–660 DOI 10.1023/A:1014697919230 | MR 1868897 | Zbl 1078.62525
[16] Takahashi R., Sibuya M.: Maximum size prediction in Wicksell’s corpuscle problem for the exponential tail data. Extremes 5 (2002), 55–70 DOI 10.1023/A:1020982025786 | MR 1947788 | Zbl 1037.62098
[17] Weissman I.: Estimation of parameters and large quantiles based on the $k$ largest observations. J. Amer. Statist. Assoc. 73 (1978), 812–815 MR 0521329 | Zbl 0397.62034
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