# zbMATH — the first resource for mathematics

Extremes of spheroid shape factor based on two dimensional profiles. (English) Zbl 1249.60105
Summary: 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 distribution 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 finalyy discuss the ML estimator based on the largest observations, and hence the possible statistical applications.

##### MSC:
 60G70 Extreme value theory; extremal stochastic processes 62G32 Statistics of extreme values; tail inference 62P30 Applications of statistics in engineering and industry; control charts
Full Text:
##### References:
 [1] Beneš V., Bodlák, K., Hlubinka D.: Stereology of extremes; FGM bivariate distributions. Method. Comput. Appl. Probab. 5 (2003), 289-308 · Zbl 1041.62039 · doi:10.1023/A:1026283103180 [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. Probability Theory and Applications (J. Galambos and J. Kátai, Kluwer, Dordrecht 1992, pp. 205-220 · Zbl 0767.60008 [5] Embrechts P., Klüppelberg, C., Mikosch T.: Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin 1997 · 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 · Zbl 0226.60039 [7] Hlubinka D.: Stereology of extremes; shape factor of spheroids. Extremes 5 (2003), 5-24 · Zbl 1051.60011 · doi:10.1023/A:1026234329084 [8] Hlubinka D.: Stereology of extremes; size of spheroids. Mathematica Bohemica 128 (2003), 419-438 · Zbl 1053.60053 · eudml:232264 [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 · 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 · 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 · Zbl 0617.62050 · doi:10.1016/0167-7152(87)90039-3 [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 · Zbl 0864.60010 · doi:10.1007/BF00049294 [14] Takahashi R., Sibuya M.: Prediction of the maximum size in Wicksell’s corpuscle problem. Ann. Inst. Statist. Math. 50 (1998), 361-377 · Zbl 0986.62075 · doi:10.1023/A:1003451417655 [15] Takahashi R., Sibuya M.: Prediction of the maximum size in Wicksell’s corpuscle problem. II. Ann. Inst. Statist. Math. 53 (2001), 647-660 · Zbl 1078.62525 · doi:10.1023/A:1014697919230 [16] Takahashi R., Sibuya M.: Maximum size prediction in Wicksell’s corpuscle problem for the exponential tail data. Extremes 5 (2002), 55-70 · Zbl 1037.62098 · doi:10.1023/A:1020982025786 [17] Weissman I.: Estimation of parameters and large quantiles based on the $$k$$ largest observations. J. Amer. Statist. Assoc. 73 (1978), 812-815 · Zbl 0397.62034 · doi:10.2307/2286285
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.