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Stereology of Extremes; Bivariate Models and Computation

Author

Listed:
  • Viktor Beneš

    (Charles University
    Academy of Sciences of the Czech Republic)

  • Karel Bodlák

    (Charles University)

  • Daniel Hlubinka

    (Charles University)

Abstract

The extremal shape factor of spheroidal particles is studied in the stereological context using the extreme value theory. The domain of attraction is invariant with respect to the transformation between spatial characteristics and planar sections characteristics. It is shown that for the Farlie-Gumbel-Morgenstern bivariate distribution of size and shape factor one can estimate the normalizing constants of shape factor conditioned by unknown particle size. The theoretical solution is followed by a detailed simulation study which demonstrates the use of estimation techniques developed. The method is useful for engineering applications in materials science, where microstructural extremes correlate with the properties of materials.

Suggested Citation

  • Viktor Beneš & Karel Bodlák & Daniel Hlubinka, 2003. "Stereology of Extremes; Bivariate Models and Computation," Methodology and Computing in Applied Probability, Springer, vol. 5(3), pages 289-308, September.
    • Handle: RePEc:spr:metcap:v:5:y:2003:i:3:d:10.1023_a:1026283103180
    DOI: 10.1023/A:1026283103180
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    References listed on IDEAS

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    1. Rinya Takahashi & Masaaki Sibuya, 1996. "The maximum size of the planar sections of random spheres and its application to metallurgy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(1), pages 127-144, March.
    2. Takahashi, Rinya, 1987. "Normalizing constants of a distribution which belongs to the domain of attraction of the Gumbel distribution," Statistics & Probability Letters, Elsevier, vol. 5(3), pages 197-200, April.
    3. Rinya Takahashi & Masaaki Sibuya, 1998. "Prediction of the Maximum Size in Wicksell's Corpuscle Problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(2), pages 361-377, June.
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