Geodesic
Stereology of extremes; size of spheroids
Mathematica Bohemica, Tome 128 (2003) no. 4, pp. 419-438

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MR Zbl
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.
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.
DOI : 10.21136/MB.2003.134007
Classification : 60G70, 62G32, 62P30
Keywords: sample extremes; domain of attraction; normalizing constants; FGM system of distributions 
Hlubinka, Daniel. Stereology of extremes; size of spheroids. Mathematica Bohemica, Tome 128 (2003) no. 4, pp. 419-438. doi: 10.21136/MB.2003.134007
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[1] V. Beneš, K. Bodlák, D. Hlubinka: Stereology of extremes; bivariate models and computation. Methodol. Comput. Appl. Probab 5 (2003), no. 3, 289–308. | DOI | MR

[2] L.-M. Cruz-Orive: Particle size-shape distributions; the general spheroid problem. J. Microscopy 107 (1976), no. 3, 235–253. | DOI

[3] H. Drees, R.-D. Reiss: Tail behavior in Wicksell’s corpuscle problem. Probability Theory and Applications, J. Galambos, J. Kátai (eds.), Kluwer, Dordrecht, 1992, pp. 205–220. | MR

[4] P. Embrechts, C. Klüppelberg, T. Mikosh: Modelling Extremal Events. Springer, Berlin, 1997. | MR

[5] L. de Haan: On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Math. Centre Tracts 32, Mathematisch Centrum, Amsterdam, 1970. | MR | Zbl

[6] B. M. Hill: A simple general approach to inference about the tail of a distribution. Ann. Stat. (1975), 1163–1174. | MR | Zbl

[7] D. Hlubinka: Stereology of extremes; shape factor of spheroids. Extremes 6 (2003), no. 1, 5–24. | DOI | MR | Zbl

[8] D. Hlubinka: Extremes of spheroid shape factor based on two dimensional profiles. (2003) (to appear). | MR

[9] R.-D. Reiss: A Course on Point Processes. Springer, New York, 1993. | MR | Zbl

[10] R.-D. Reiss, M. Thomas: Statistical Analysis of Extreme Values. From Insurance, Finance, Hydrology and Other Fields. Birkhäuser, Basel, 2001. | MR

[11] R. Takahashi: Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution. Stat. Probab. Lett. 5 (1987), 197–200. | DOI | MR | Zbl

[12] R. Takahashi, M. Sibuya: The maximum size of the planar sections of random spheres and its application to metalurgy. Ann. Inst. Stat. Math. 48 (1996), no. 1, 127–144. | DOI | MR

[13] R. Takahashi, M. Sibuya: Prediction of the maximum size in Wicksell’s corpuscle problem. Ann. Inst. Stat. Math. 50 (1998), no. 2, 361–377. | DOI | MR

[14] R. Takahashi, M. Sibuya: Prediction of the maximum size in Wicksell’s corpuscle problem. Ann. Inst. Stat. Math. 53 (2001), no. 3, 647–660. | DOI | MR

[15] I. Weissman: Estimation of parameters and large quantiles based on the $k$ largest observations. J. Am. Stat. Assoc. 73 (1978), no. 364, 812–815. | MR | Zbl

[16] S. D. Wicksell: The corpuscle problem I. Biometrika 17 (1925), 84–99.

[17] S. D. Wicksell: The corpuscle problem II. Biometrika 18 (1926), 152–172.

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