Mathematical Geology

, Volume 21, Issue 8, pp 829–842 | Cite as

On tail estimation: An improved method

  • G. R. Dargahi-Noubary
Articles

Abstract

A step is described toward better statistical treatment of data for tail estimation. The classical extreme value theory together with its practical inefficiency for tail inference are discussed briefly. The threshold method that utilizes available information in a more efficient manner is described, and its relation to extreme value theory is mentioned. Some comparison is also made using two sets of published data.

Key words

Tail extremes threshold generalized pareto wind flood 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bardslley, W. E., 1988, Toward a General Procedure for Analysis of Extreme Random Events in the Earth Sciences: Math. Geol., v. 20, n. 5, p. 513–528.Google Scholar
  2. Benjamin, J. R., and Cornell, C. A., 1970, Probability, Statistics and Decision for Civil Engineers, McGraw-Hill, New York, 684 p.Google Scholar
  3. Cook, N. J., 1982, Towards Better Estimation of Extreme Winds: J. Wind Eng. Ind. Aerodyn., v. 9, p. 295–323.Google Scholar
  4. Cox, D. R., and Lewis, P. A. W., 1966, The Statistical Analysis of Series of Events, Chapman and Hall, London, 285 p.Google Scholar
  5. David, H. A., 1981, Order Statistics, John Wiley, New York, 360 p.Google Scholar
  6. Davison, A., 1984, Modelling Excesses over High Thresholds, With an Application. In Statistical Extremes and Applications, in J. Tiago de Oliveira (Ed.), D. Reidel, Dordrecht, p. 461–482.Google Scholar
  7. Dekkers, A. L. M., and De Haan, L., 1987, On a Consistent Estimate of the Index of an Extreme Value Distribution, Report MS-R8710, Center for Math. and Comp. Science, Amsterdam, The Netherlands, 17 p.Google Scholar
  8. Dorman, C. M. L., 1982, Extreme Values: An Improved Method of Fitting: J. Wind Eng. Ind. Aerodyn., v. 10, p. 177–190.Google Scholar
  9. DuMouchel, W., 1983, Estimating the Stable Index α in Order to Measure Tail Thickness: Ann. Statist., v. 11, p. 1019–1036.Google Scholar
  10. Fisher, R. A., and Tippett, L. H. C., 1928, Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample: Proc. Cambridge Philos. Soc., v. 24, p. 180–190.Google Scholar
  11. Galambos, J., 1987, The Asymptotic Theory of Extreme Order Statistics: Wiley, New York, 2nd ed., 352 p.Google Scholar
  12. Gumbel, E. J., 1958, Statistics of Extremes, Columbia University Press, New York, 375 p.Google Scholar
  13. Hall, P., 1982, On Estimating the Endpoint of a Distribution: Ann. Statist., v. 10, p. 556–568.Google Scholar
  14. Hill, B. M., 1975, A Simple General Approach to Inference about the Tail of a Distribution: Ann. Statist., v. 3, p. 1163–1174.Google Scholar
  15. Hinkley, D. V., 1969, On the Ratio of Two Correlated Normal Variables: Biometrika, v. 56, p. 635–640.Google Scholar
  16. Hogg, R. V., and Tanis, E. A., 1983, Probability and Statistical Inference, 2nd ed. Macmillan, New York, 533 p.Google Scholar
  17. Johnson, N. L., and Kotz, S., 1972, Distributions in Statistics (Continuous Multivariate Distributions). John Wiley & Sons, New York, 322 p.Google Scholar
  18. Leadbetter, M. R., Lindgren, G., and Rootzen, H., 1983, Extremes and Related Properties of Random Sequences and Series, Springer-Verlag, New York, 336 p.Google Scholar
  19. NERC, The Flood Studies Report, 1975, National Environment Research Council, London, v. 1–5, Bibliography, v. 1, p. 483–485.Google Scholar
  20. North, M., 1980, Time-Dependent Stochastic Models of Floods: J. Hyd. Div. ASCE, v. 106, p. 649–655.Google Scholar
  21. Pericchi, L. R., and Rodrigues-Iturbe, I., 1985, On Statistical Analysis of Floods, in A. C. Atkinson and S. E. Fienberg (eds.) A Celebration of Statistics, The ISI Centenary Volume, Springer-Verlag, New York, p. 521–523.Google Scholar
  22. Pickands, J., 1975, Statistical Inference Using Extreme Order Statistics: Ann. Statist., v. 3, p. 119–131.Google Scholar
  23. Pourahmadi, M., 1989, Estimation and Interpolation of Missing Values of a Stationary Time Series: J. Time Series Analysis: v. 10, p. 1–21.Google Scholar
  24. Prescott, P., and Walden, A. T., 1983, Maximum likelihood estimation of the Three-Parameter Generalized Extreme-Value Distribution from Censored Samples: J. Statist. Comput. Simul., v. 16, p. 241.Google Scholar
  25. Rijkoort, P. J., and Wierings, J., 1983, Extreme Wind Speeds by Compound Weibull Analysis of Exposure-Corrected Data: J. Wind. Eng. Ind. Aerodyn., v. 13, p. 93–104.Google Scholar
  26. Schuster, E. F., 1984, Classification of Probability Laws by Tail Behavior: J. Amer. Stat. Assoc., v. 79, p. 936–939.Google Scholar
  27. Smith, R. L., 1987, Estimating Tails of Probability Distributions: Ann. Statist., v. 15, p. 1174–1207.Google Scholar
  28. Todorovic, P., 1979, A Probabilistic Approach to Analysis and Prediction of Floods: Proc. of the 42nd Section of the ISI, International Statistical Institute, Buenos Aires, p. 113–124.Google Scholar
  29. Von Mises, R., 1936, La Distribution de la plus Grande n Valeurs, in Selected Papers II, American Mathematical Society, Providence, p. 271–294.Google Scholar
  30. Weibull, W., 1939, A Statistical Theory of Strength of Material, Ingeniors Vetenskaps Akademien handlingar, n. 151, 139 p.Google Scholar

Copyright information

© International Association for Mathematical Geology 1989

Authors and Affiliations

  • G. R. Dargahi-Noubary
    • 1
  1. 1.Department of Mathematical Sciences, Division of StatisticsUniversity of Northern IllinoisDeKalb
  2. 2.Department of Mathematics and Computer ScienceBloomsburg UniversityBloomsburg

Personalised recommendations