Mathematical Geology

, Volume 28, Issue 1, pp 25–43 | Cite as

Extreme value analysis of diamond-size distributions

  • J. Caers
  • P. Vynckier
  • J. Beirlant
  • L. Rombouts


Extreme value analysis provides a semiparametric method for analyzing the extreme long tails of skew distributions which may be observed when handling mining data. The estimation of important tail characteristics, such as the extreme value index, allows for a discrimination between competing distribution models. It measures the “thickness” of such long tailed distributions, if only a limited sample is available. This paper stresses the practical implementation of extreme value theory, which is used to discriminate a lognormal from a mixed lognormal distribution in a case study of size distributions for alluvial diamonds.

Key words

extreme value theory quantile-quantile plot loghyperbolic lognormal diamond 


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  1. Aitchison, J., and Brown, J. A. C. 1969, The lognormal distribution with special references to its uses in economics: Cambridge Univ. Press, Cambridge, 176 p.Google Scholar
  2. Bardsley, W. E., 1988, Towards a general procedure for the analysis of extreme random events in the earth sciences: Math. Geology, v. 20, no. 5, p. 513–528.CrossRefGoogle Scholar
  3. Barndorff-Nielsen, O., 1977, Exponentially decreasing distributions for the logarithm of particle size: Proc. Roy. Soc. London, Ser. A. 353, no. 1674, p. 401–419.Google Scholar
  4. Barndorff-Nielsen, O., and Christiansen, C., 1988, Erosion, deposition and size distributions of sand: Proc. Roy. Soc. London, Ser. A 417, no. 1853, p. 335–352.Google Scholar
  5. Barndorff-Nielsen, O., and Sørensen, M., 1981, On the temporal spatial variation of sediment size distributions: Acta Mech. Suppl., v. 2, no. 1, p. 23–35.Google Scholar
  6. Barndorff-Nielsen, O., Kent, J., and Sørensen, M., 1982, Normal variance-mean mixture and z-distributions: Intern. Stat. Rev., v. 50, no. 3, p. 145–159.Google Scholar
  7. Beirlant. J., Vynckier, P., and Teugels, J. L., 1995a, Tail index estimation, pareto quantile plots and regression diagnostics: publication submitted to the Jour. Am. Stat. Assoc.Google Scholar
  8. Beirlant, J., Vynckier, P., and Teugels, J. L., 1995b, Excess functions and estimation of the extreme-value index: publication submitted to Bernoulli.Google Scholar
  9. Caers, J., and Vynckier, P., 1995, Quantile-Quantile plots and extreme value statistics, indicators of truly non-Gaussian variation: Cahiers de Géostatistique, Centre de Géostatistique, fasc. 5, Fontainebleau, in press.Google Scholar
  10. Csörgő, S., Deheuvels, P., and Mason, D., 1985, Kernel estimates of the tail index of a distribution: Ann. Stat., v. 13, no. 3, p. 1050–1077.Google Scholar
  11. Dargahi-Noubary, G. R., 1989, On tail estimation: an improved method: Math Geology, v. 21, no. 8, p. 829–842.CrossRefGoogle Scholar
  12. de Haan, L., 1970, On regular variation and its applications to the weak convergence of sample extremes: Math. Centre, Tract 32, Amsterdam, unpaginated.Google Scholar
  13. Deigaard, R., and Fredsøe, J., 1978, A sediment transport model for straight alluvial channels: Nordic. Hydrol., v. 7, no. 4, p. 293–309.Google Scholar
  14. Dekkers, L. M., Einmahl, J. H. J., and de Haan, L., 1989, A momentestimator for the index of the extreme value distribution: Ann. Stat., v. 17, no. 4, p. 1833–1855.Google Scholar
  15. Fisher, R. A., and Tippett, L. H. C., 1928, Limiting forms of the frequency distribution in the largest particle size and smallest member of a sample: Proc. Cambridge Phil. Soc, v. 24, p. 180–190.Google Scholar
  16. Gnedenko, B. V., 1943, Sur la distribution limite du terme maximum d'une variable aléatoire: Ann. Math., v. 44, no. 3, p. 423–453.Google Scholar
  17. Gumbel, E. J., 1958, Statistics of extremes: Columbia Univ. Press, New York, 375 p.Google Scholar
  18. Hill, B. M., 1975, A simple and general approach to inference about the tail of a distribution: Ann. Stat., v. 3. no. 5. p. 1163–1174.Google Scholar
  19. Houghton, J. C., 1988, Use of the truncated shifted pareto distribution in assessing size distribution of oil and gas fields: Math. Geology, v. 20, no. 8, p. 907–937.CrossRefGoogle Scholar
  20. Johnson, N. L., and Kotz, S., 1970, Continuous univariate distributions-1: Houghton Mifflin, Boston, 300 p.Google Scholar
  21. Krige, D. G., and Dohm, C. E., 1994, The role of massive grade data bases in geostatistical applications in South African gold mines,in Dimitrakopaulos, R., ed., Geostatistics for the next century: Kluwer Acad., Dordrecht and Boston, 497 p.Google Scholar
  22. Leadbetter, M. R., and Roótzen, H., 1988, Extremal theory for stochastic processes: Ann. Prob., v. 16, no. 2, p. 431–478.Google Scholar
  23. Matheron, G., 1989, Estimating and choosing: Springer Verlag, Berlin, 141 p.Google Scholar
  24. Pickands, J., 1975, Statistical inference using extreme order statistics: Ann. Stat., v. 3, no. 1, p. 119–131.Google Scholar
  25. Rombouts, L., 1987, Geology and evaluation of the Guinean diamond deposits: Ann. Soc. Géol. Belg., v. 110, p. 241–259.Google Scholar
  26. Seyedghasemipour, S. J., and Bhattacharyya, B. B., 1990, The loghyperbolic: an alternative to the lognormal for modeling oil field size distribution: Math. Geology, v. 22, no. 5, p. 557–571.CrossRefGoogle Scholar
  27. Sichel, H. S., 1966, The estimations of the means and associated confidence limits for small samples from lognormal populations. Symposium on mathematical statistics and computer applications in ore valuation: Jour. South Afr. Inst. Min. Met., v. 66, p. 106–122.Google Scholar
  28. Sichel, H. S., 1973, Statistical valuation of diamondiferous deposits. Jour. South Afr. Inst. Min. Met., v. 73, p. 235–243.Google Scholar
  29. Sichel, H. S., 1987, Some advances in lognormal theory: APCOM 87, v. 3: Geostatistics. South Afr. Inst. Min. Met., Johannesburg, p. 3–8.Google Scholar
  30. Sichel, H. S., Kleingeld, W. J., and Assibey-Bonsu, W., 1992, A comparative study of three frequency-distribution models for use in ore evaluation: Jour. South Afr. Inst. Min. Met., v. 92. p. 235–243.Google Scholar
  31. Sichel, H. S., Dohm, C. E., and Kleingeld, W. J., 1995, New generalized model of observed ore value distributions; Trans. Instn. Min. Metall. (sect. A: Min. industry), v. 104, no. 2, p. 116–123.Google Scholar
  32. Smith, R. L., 1987. Estimating tails of probability distributions: Ann. Stat., v. 15, no. 3, p. 1174–1207.Google Scholar
  33. Smith, R. L., 1989, Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone: Statistical Science, v. 4, no. 4, p. 367–393.MathSciNetGoogle Scholar

Copyright information

© International Association for Mathematical Geology 1996

Authors and Affiliations

  • J. Caers
    • 1
  • P. Vynckier
    • 2
  • J. Beirlant
    • 3
  • L. Rombouts
    • 4
  1. 1.Department of Civil EngineeringKULeuvenHeverleeBelgium
  2. 2.Research Assistant of the Belgian National Fund for Scientific Research (NFWO)Belgium
  3. 3.Department of MathematicsKULeuvenHeverleeBelgium
  4. 4.Terraconsult bvbaMortselBelgium

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