Stereology of extremes; size of spheroids

Daniel Hlubinka

Mathematica Bohemica (2003)

  • Volume: 128, Issue: 4, page 419-438
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Hlubinka, Daniel. "Stereology of extremes; size of spheroids." Mathematica Bohemica 128.4 (2003): 419-438. <http://eudml.org/doc/249211>.

@article{Hlubinka2003,
abstract = {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.},
author = {Hlubinka, Daniel},
journal = {Mathematica Bohemica},
keywords = {sample extremes; domain of attraction; normalizing constants; FGM system of distributions; sample extremes; domain of attraction; normalizing constants; FGM system of distributions},
language = {eng},
number = {4},
pages = {419-438},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stereology of extremes; size of spheroids},
url = {http://eudml.org/doc/249211},
volume = {128},
year = {2003},
}

TY - JOUR
AU - Hlubinka, Daniel
TI - Stereology of extremes; size of spheroids
JO - Mathematica Bohemica
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 128
IS - 4
SP - 419
EP - 438
AB - 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.
LA - eng
KW - sample extremes; domain of attraction; normalizing constants; FGM system of distributions; sample extremes; domain of attraction; normalizing constants; FGM system of distributions
UR - http://eudml.org/doc/249211
ER -

References

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  1. 10.1023/A:1026283103180, Methodol. Comput. Appl. Probab 5 (2003), no. 3, 289–308. (2003) MR2016768DOI10.1023/A:1026283103180
  2. 10.1111/j.1365-2818.1976.tb02446.x, J. Microscopy 107 (1976), no. 3, 235–253. (1976) DOI10.1111/j.1365-2818.1976.tb02446.x
  3. Tail behavior in Wicksell’s corpuscle problem, Probability Theory and Applications, J. Galambos, J. Kátai (eds.), Kluwer, Dordrecht, 1992, pp. 205–220. (1992) MR1211909
  4. Modelling Extremal Events, Springer, Berlin, 1997. (1997) MR1458613
  5. On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, Math. Centre Tracts 32, Mathematisch Centrum, Amsterdam, 1970. (1970) Zbl0226.60039MR0286156
  6. A simple general approach to inference about the tail of a distribution, Ann. Stat. (1975), 1163–1174. (1975) Zbl0323.62033MR0378204
  7. 10.1023/A:1026234329084, Extremes 6 (2003), no. 1, 5–24. (2003) Zbl1051.60011MR2021590DOI10.1023/A:1026234329084
  8. Extremes of spheroid shape factor based on two dimensional profiles, (2003) (to appear). (ARRAY(0x8a4d390)) MR2021590
  9. A Course on Point Processes, Springer, New York, 1993. (1993) Zbl0771.60037MR1199815
  10. Statistical Analysis of Extreme Values. From Insurance, Finance, Hydrology and Other Fields, Birkhäuser, Basel, 2001. (2001) MR1819648
  11. 10.1016/0167-7152(87)90039-3, Stat. Probab. Lett. 5 (1987), 197–200. (1987) Zbl0617.62050MR0881196DOI10.1016/0167-7152(87)90039-3
  12. 10.1007/BF00049294, Ann. Inst. Stat. Math. 48 (1996), no. 1, 127–144. (1996) MR1392521DOI10.1007/BF00049294
  13. 10.1023/A:1003451417655, Ann. Inst. Stat. Math. 50 (1998), no. 2, 361–377. (1998) MR1868939DOI10.1023/A:1003451417655
  14. 10.1023/A:1014697919230, Ann. Inst. Stat. Math. 53 (2001), no. 3, 647–660. (2001) MR1868897DOI10.1023/A:1014697919230
  15. Estimation of parameters and large quantiles based on the k largest observations, J. Am. Stat. Assoc. 73 (1978), no. 364, 812–815. (1978) Zbl0397.62034MR0521329
  16. The corpuscle problem I, Biometrika 17 (1925), 84–99. (1925) 
  17. The corpuscle problem II, Biometrika 18 (1926), 152–172. (1926) 

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