Previous |  Up |  Next

Article

Title: The generalized FGM distribution and its application to stereology of extremes (English)
Author: Hlubinka, Daniel
Author: Kotz, Samuel
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 55
Issue: 6
Year: 2010
Pages: 495-512
Summary lang: English
.
Category: math
.
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. (English)
Keyword: generalized FGM distribution
Keyword: extremes
Keyword: stereology
Keyword: maximum domain of attraction
MSC: 60G70
MSC: 62G32
MSC: 62H05
MSC: 62H10
MSC: 62P99
idZBL: Zbl 1223.62080
idMR: MR2737716
DOI: 10.1007/s10492-010-0020-x
.
Date available: 2010-11-24T08:21:25Z
Last updated: 2020-07-02
Stable URL: http://hdl.handle.net/10338.dmlcz/140823
.
Reference: [1] Bairamov, I., Kotz, S.: Dependence structure and symmetry of Huang-Kotz {FGM} distributions and their extensions.Metrika 56 (2002), 55-72. MR 1922211, 10.1007/s001840100158
Reference: [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. MR 1836732, 10.1080/02664760120047861
Reference: [3] Bairamov, I., Kotz, S., Gebizlioglu, O. L.: The Sarmanov family and its generalization.S. Afr. Stat. J. 35 (2001), 205-224. Zbl 1009.62011, MR 1910896
Reference: [4] Beneš, V., Bodlák, K., Hlubinka, D.: Stereology of extremes; bivariate models and computation.Methodol. Comput. Appl. Probab. 5 (2003), 289-308. Zbl 1041.62039, MR 2016768, 10.1023/A:1026283103180
Reference: [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. 10.1016/S1359-6454(96)00249-2
Reference: [6] Coles, S.: An Introduction to Statistical Modeling of Extreme Values.Springer London (2001). Zbl 0980.62043, MR 1932132
Reference: [7] Cruz-Orive, L. M.: Particle size-shape distributions; the general spheroid problem.J. Microscopy 107 (1976), 235-253. 10.1111/j.1365-2818.1976.tb02446.x
Reference: [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. Zbl 0767.60008, MR 1211909
Reference: [9] Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Applications of Mathematics. 33.Springer Berlin (1997). MR 1458613
Reference: [10] Gnedenko, B. V.: Sur la distribution limite du terme maximum d'une série aléatoire.Ann. Math. 44 (1943), 423-453. Zbl 0063.01643, MR 0008655, 10.2307/1968974
Reference: [11] Hlubinka, D.: Extremes of spheroid shape factor based on two dimensional profiles.Kybernetika (to appear). MR 2208521
Reference: [12] Hlubinka, D.: Stereology of extremes; shape factor of spheroids.Extremes 6 (2003), 5-24. Zbl 1051.60011, MR 2021590, 10.1023/A:1026234329084
Reference: [13] Hlubinka, D.: Stereology of extremes; size of spheroids.Math. Bohem. 128 (2003), 419-438. Zbl 1053.60053, MR 2032479
Reference: [14] Hlubinka, D.: Bivariate models for extremes in stereology.Tech. Rep., KPMS, Charles University in Prague (2005). MR 2021590
Reference: [15] Johnson, N. L., Kotz, S.: On some generalized Farlie-Gumbel-Morgenstern distributions.Commun. Stat. 4 (1975), 415-427. Zbl 0342.62006, MR 0373155, 10.1080/03610927508827258
Reference: [16] Nelsen, R. B.: An Introduction to Copulas. Lecture Notes in Statistics, No 139.Springer New York (1999). MR 1653203, 10.1007/978-1-4757-3076-0_1
Reference: [17] Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures in Materials Science.Wiley Chichester (2000).
Reference: [18] Rodríguez-Lallena, J. A., Úbeda-Flores, M.: A new class of bivariate copulas.Stat. Probab. Lett. 66 (2004), 315-325. Zbl 1102.62054, MR 2045476, 10.1016/j.spl.2003.09.010
Reference: [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. Zbl 0864.60010, MR 1392521, 10.1007/BF00049294
Reference: [20] Takahashi, R., Sibuya, M.: Prediction of the maximum size in Wicksell's corpuscle problem.Ann. Inst. Stat. Math. 50 (1998), 361-377. Zbl 0986.62075, MR 1868939, 10.1023/A:1003451417655
Reference: [21] Takahashi, R., Sibuya, M.: Prediction of the maximum size in Wicksell's corpuscle problem. II.Ann. Inst. Stat. Math. 53 (2001), 647-660. Zbl 1078.62525, MR 1868897, 10.1023/A:1014697919230
Reference: [22] Takahashi, R., Sibuya, M.: Maximum size prediction in Wicksell's corpuscle problem for the exponential data.Extremes 5 (2002), 55-70. MR 1947788, 10.1023/A:1020982025786
Reference: [23] Takahashi, R., Sibuya, M.: Metal fatigue, Wicksell transform and extreme values.Appl. Stoch. Models Bus. Ind. 18 (2002), 301-312. Zbl 1010.62113, MR 1932643, 10.1002/asmb.477
Reference: [24] Weissman, I.: Estimation of parameters and large quantiles based on the $k$ largest observations.J. Am. Stat. Assoc. 73 (1978), 812-815. Zbl 0397.62034, MR 0521329
Reference: [25] Wicksell, S. D.: The corpuscle problem I. A mathematical study of a biometric problem.Biometrika 17 (1925), 84-99.
Reference: [26] Wicksell, S. D.: The corpuscle problem II.Biometrika 18 (1926), 152-172.
.

Files

Files Size Format View
AplMat_55-2010-6_3.pdf 308.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo