Article
MSC:
60G70,
62G32,
62H05,
62H10,
62P99 | MR 2737716 | Zbl 1223.62080 | DOI: 10.1007/s10492-010-0020-x
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Keywords:
generalized FGM distribution; extremes; stereology; maximum domain of attraction
generalized FGM distribution; extremes; stereology; maximum domain of attraction
Summary:
The generalized FGM distribution and related copulas are used as bivariate models for the distribution of spheroidal characteristics. It is shown that this model is suitable for the study of extremes of the 3D spheroidal particles observed in terms of their random planar sections.
References:
[1] Bairamov, I., Kotz, S.: Dependence structure and symmetry of Huang-Kotz {FGM} distributions and their extensions. Metrika 56 (2002), 55-72. DOI 10.1007/s001840100158 | MR 1922211
[2] Bairamov, I., Kotz, S., Bekçi, M.: New generalized Farlie-Gumbel-Morgenstern distribution and concomitants of order statistics. J. Appl. Stat. 28 (2001), 521-536. DOI 10.1080/02664760120047861 | MR 1836732
[3] Bairamov, I., Kotz, S., Gebizlioglu, O. L.: The Sarmanov family and its generalization. S. Afr. Stat. J. 35 (2001), 205-224. MR 1910896 | Zbl 1009.62011
[4] Beneš, V., Bodlák, K., Hlubinka, D.: Stereology of extremes; bivariate models and computation. Methodol. Comput. Appl. Probab. 5 (2003), 289-308. DOI 10.1023/A:1026283103180 | MR 2016768 | Zbl 1041.62039
[5] 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
[6] Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer London (2001). MR 1932132 | Zbl 0980.62043
[7] Cruz-Orive, L. M.: Particle size-shape distributions; the general spheroid problem. J. Microscopy 107 (1976), 235-253. DOI 10.1111/j.1365-2818.1976.tb02446.x
[8] Dress, H., Reiss, R.-D.: Tail behavior in Wicksell's corpuscle problem. In: Probability Theory and Applications J. Galambos, J. Kátai Kluwer Dordrecht (1992), 205-220. MR 1211909 | Zbl 0767.60008
[9] Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Applications of Mathematics. 33. Springer Berlin (1997). MR 1458613
[10] Gnedenko, B. V.: Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44 (1943), 423-453. DOI 10.2307/1968974 | MR 0008655 | Zbl 0063.01643
[11] Hlubinka, D.: Extremes of spheroid shape factor based on two dimensional profiles. Kybernetika (to appear). MR 2208521
[12] Hlubinka, D.: Stereology of extremes; shape factor of spheroids. Extremes 6 (2003), 5-24. DOI 10.1023/A:1026234329084 | MR 2021590 | Zbl 1051.60011
[13] Hlubinka, D.: Stereology of extremes; size of spheroids. Math. Bohem. 128 (2003), 419-438. MR 2032479 | Zbl 1053.60053
[14] Hlubinka, D.: Bivariate models for extremes in stereology. Tech. Rep., KPMS, Charles University in Prague (2005). MR 2021590
[15] Johnson, N. L., Kotz, S.: On some generalized Farlie-Gumbel-Morgenstern distributions. Commun. Stat. 4 (1975), 415-427. DOI 10.1080/03610927508827258 | MR 0373155 | Zbl 0342.62006
[16] Nelsen, R. B.: An Introduction to Copulas. Lecture Notes in Statistics, No 139. Springer New York (1999). DOI 10.1007/978-1-4757-3076-0_1 | MR 1653203
[17] Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures in Materials Science. Wiley Chichester (2000).
[18] Rodríguez-Lallena, J. A., Úbeda-Flores, M.: A new class of bivariate copulas. Stat. Probab. Lett. 66 (2004), 315-325. DOI 10.1016/j.spl.2003.09.010 | MR 2045476 | Zbl 1102.62054
[19] Takahashi, R., Sibuya, M.: The maximum size of the planar sections of random spheres and its application to metallurgy. Ann. Inst. Stat. Math. 48 (1996), 127-144. DOI 10.1007/BF00049294 | MR 1392521 | Zbl 0864.60010
[20] Takahashi, R., Sibuya, M.: Prediction of the maximum size in Wicksell's corpuscle problem. Ann. Inst. Stat. Math. 50 (1998), 361-377. DOI 10.1023/A:1003451417655 | MR 1868939 | Zbl 0986.62075
[21] Takahashi, R., Sibuya, M.: Prediction of the maximum size in Wicksell's corpuscle problem. II. Ann. Inst. Stat. Math. 53 (2001), 647-660. DOI 10.1023/A:1014697919230 | MR 1868897 | Zbl 1078.62525
[22] Takahashi, R., Sibuya, M.: Maximum size prediction in Wicksell's corpuscle problem for the exponential data. Extremes 5 (2002), 55-70. DOI 10.1023/A:1020982025786 | MR 1947788
[23] Takahashi, R., Sibuya, M.: Metal fatigue, Wicksell transform and extreme values. Appl. Stoch. Models Bus. Ind. 18 (2002), 301-312. DOI 10.1002/asmb.477 | MR 1932643 | Zbl 1010.62113
[24] Weissman, I.: Estimation of parameters and large quantiles based on the $k$ largest observations. J. Am. Stat. Assoc. 73 (1978), 812-815. MR 0521329 | Zbl 0397.62034
[25] Wicksell, S. D.: The corpuscle problem I. A mathematical study of a biometric problem. Biometrika 17 (1925), 84-99.
[26] Wicksell, S. D.: The corpuscle problem II. Biometrika 18 (1926), 152-172.
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