Skip to main content
SpringerLink
Log in
Book cover

Entropy-Based Parameter Estimation in Hydrology pp 108–136Cite as

Extreme Value Type 1 Distribution

  • Chapter

Part of the book series: Water Science and Technology Library ((WSTL,volume 30))

Abstract

The extreme value type 1 (EV 1) distribution is one of the most popularly used distributions for frequency analysis of extreme values of meteorologic or climatic and hydrologic variables, such as floods, rainfall, droughts, etc. This distribution was derived by Fisher and Tippett (1928) as a limiting form of the frequency distribution of the largest or smallest of a sample. In a series of papers Gumbel (1941a, b, 1942a, b, 1948) derived the EV1 distribution for flood flows and applied it to frequency analysis of floods, droughts, and meteorological data. Gumbel (1958) published a treatise on statistics of extremes, which contains a comprehensive treatment of EV1 distribution. Bardsley and Manly (1987) examined the transformations under which non-Gumbel distributions of annual flood flow maxima would converge to the Gumbel distribution. Smith (1986) presented a family of statistical distributions and estimators based on a fixed number (greater than one) of the largest annual events. Jenkinson (1955) found a general solution of the function equation derived by Fisher and Tippett (1928) for extreme values and showed that the Gumbel distribution was a special case of the general solution. Singh et al. (1986) derived this distribution using the principle of maximum entropy. Al-Mashidini et al. (1978) presented a simplified form of EV 1 distribution for flood estimation.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Adamowski, K., 1981. Plotting formula for flood frequency. Water Resources Bulletin, Vol. 17, No. 2, pp. 197–202.

    Article  Google Scholar 

  • Al-Mashidini, G., Pande, B.B.L. and Mujda, M.F., 1978. A simple version of Gumbel’s method for flood estimation. Hydrological Sciences Bulletin, Vol. 23, No. 3, pp. 373–380.

    Article  Google Scholar 

  • Anderson, R.L., 1941. Distribution of the serial correlation coefficient. Annals of Mathematical Statistics, Vol. 8, pp. 1–13.

    Google Scholar 

  • Arora, K. and Singh, V.P., 1987. On statistical intercomparison of EV1 estimators by Monte Carlo simulation. Advances in Water Resources, Vol. 10, No. 2, pp. 87–107.

    Article  Google Scholar 

  • Bardsley, W.E. and Manly, B.F.J., 1987. Transformations for improved convergence of distribution of flood maxima to a Gumbel limit. Journal of Hydrology, Vol. 91, pp. 137152.

    Google Scholar 

  • Benson, M.A., 1968. Uniform flood frequency esimating methods for federal agencies. Water Resources Research, Vol. 4, No., pp. 891–908.

    Google Scholar 

  • Bobee, B. and Robitaille, 1977. The use of Pearson type 3 and log Pearson type 3 distributions revisited. Water Resources Research, Vol. 13, No. 2, pp. 427–443.

    Article  Google Scholar 

  • Chow, V.T., 1953. Frequency analysis of hydrologic data. Bulletin 414, Engineering Experiment Station, University of Illinois, Urbana, Illinois

    Google Scholar 

  • Coulson, A., 1966. Tables for computing and plotting flood frequency curves. Technical Bulletin No. 3, Water Resources Branch, Department of Energy, Mines and Resources, Ottawa, Canada.

    Google Scholar 

  • Fill, H.D. and Stedinger, J.R., 1995. L moment and probability plot correlation coefficient goodness-of-fit tests for the Gumbel distribution and impact of autocorrelation. Water Resources Research, Vol. 31, No. 1, pp. 225–229.

    Article  Google Scholar 

  • Fisher, R.A. and Tippett, L.H.C., 1928. Limiting forms of the frequency distribution of the largest or smallest member of a sample Proceedings, Cambridge Philosophical Society, Vol. 24, pp. 48–55.

    Article  Google Scholar 

  • Gumbel, E.J., 1941 a. The return period of flood flows. Annals of Mathematical Statistics, Vol. XII, No. 2, pp. 163–190.

    Google Scholar 

  • Gumbel, 1941b. The probability-interpretation of the observed return periods of floods. Transactions, American Geophysical Union, Vol. 22, Part III, pp. 836–849.

    Google Scholar 

  • Gumbel, E.J., 1942a. On the frequency distribution of extreme values in meteorological data. Bulletin of the American Meteorological Society, Vol. 23, No. 3, pp. 95–105.

    Google Scholar 

  • Gumbel, E.J., 1942b. Statistical control-curves for flood discharges. Transactions, American Geophysical Union, Vol. 23, pp. 489–500.

    Article  Google Scholar 

  • Gumbel, E.J., 1948. The statistical forecast of floods. Technical Report, Ohio Water Resources Board, Columbus, Ohio.

    Google Scholar 

  • Gumbel, E.J., 1958. Statistics of Extremes. Columbia University Press, Columbia, New York

    Google Scholar 

  • Hersfield, D.M. and Kohler, M.A., 1960. An empirical appraisal of the Gumbel extreme value procedure. Journal of Geophysical Research, Vol. 65, No. 6, pp. 1737–1746.

    Article  Google Scholar 

  • Houghton, J.C., 1978. The incomplete means estimation procedure applied to flood frequency analysis. Water Resources Research, Vol. 14, No. 6, pp. 1111–1115

    Article  Google Scholar 

  • Jain, D. and Singh, V.P., 1987. Estimating parameters of EV1 distribution for flood frequency analysis. Bulletin, American Water Resources Association, Vol. 23, No. 1, pp. 59–71.

    Article  Google Scholar 

  • Jenkinson, A.F., 1955. The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly.Journal of Royal Meteorological Society, Vol. 81, pp. 158–171.

    Article  Google Scholar 

  • Jowitt, P.W., 1979. The extreme value type 1 distribution and the principle of maximum entropy. Journal of Hydrology, Vol. 42, pp. 23–38.

    Article  Google Scholar 

  • Kimball, B.F., 1949. An approximation to the sampling variances of an estimated maximum value of given frequency based on fit of doubly exponential distribution of maximum values. Annals of Mathematical Statistics, Vol. 20, pp. 110–113.

    Article  Google Scholar 

  • Lowery, M.D. and Nash, J.E., 1970. A comparison of methods of fitting the double exponential distribution. Journal of Hydrology, Vol. 10, pp. 259–275.

    Article  Google Scholar 

  • Lambert, J. And Duan, L., 1994. Evaluating risk of extreme events for univariate-loss functions. Journal of Water Resources Planning and Management, Vol. 120, No. 3, pp. 382–399.

    Google Scholar 

  • Landwehr, J.M., Matalas, N.C. and Wallis, J.R., 1979. Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Research, Vol. 15, No. 5, pp. 1055–1064.

    Article  Google Scholar 

  • Lattenmaier, D.P. and Burges, S.J., 1982. Gumbel’s extreme value I distribution: A new look. Journal of Hydraulics Division, ASCE, Vol. 108, No. HY4, pp. 502–514.

    Google Scholar 

  • Majumdar, K.C. and Sawhney, R.P., 1965. Estimates of extreme values by different distribution functions. Water Resources Research, Vol. 1, No. 3, pp. 429–434.

    Article  Google Scholar 

  • Mann, H.B. and Whitney, D.R., 1947. On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, Vol. 18, pp. 50–61.

    Article  Google Scholar 

  • Matalas, N.C., Slack, J.R. and Wallis, J.R., 1975. Regional skew in search of a parent. Water Resources Research, Vol. 11, No. 6, pp. 815–826.

    Article  Google Scholar 

  • Phien, H.N., 1987. A review of methods of parameter estimation fro the extreme value type-1 distribution. Journal of Hydrology, Vol. 90, pp. 251–268.

    Article  Google Scholar 

  • Phien, H.N. and Arbhabhirama, A., 1980. A comparison of statistical tests on the extreme-value type-1 distribution. Bulletin, American Water Resources Association, Vol. 16, No. 5, pp. 856–861.

    Article  Google Scholar 

  • Raynal, J.A. and Salas, J.D., 1986. Estimation procedures for the type-1 extreme value distribution. Journal of Hydrology, Vol. 87, pp. 315–336.

    Article  Google Scholar 

  • Reich, BM., 1970. Flood series compaed to rainfall extremes. Water Resources Research, Vol. 6, No. 6, pp. 1655–1667.

    Article  Google Scholar 

  • Shen, H.W., Bryson, M.C. and Ochoa, I.D., 1980. The effect of tail assumption on flood predictions. Water Resources Research, Vol. 16, pp. 361–364.

    Article  Google Scholar 

  • Siegel, S., 1956. Nonparametric Statistics for the Behavioral Sciences. Mc-Graw Hill Book Company, New York.

    Google Scholar 

  • Singh, V.P. and Rajagopal, A.K., 1986. A new method of parameter estimation for hydrologic frequency analysis. Hydrological Science and Technology, Vol. 2, No. 3, pp. 33–40.

    Google Scholar 

  • Singh, V.P., Rajagopal, A.K. and Singh, K., 1986. Derivation of some frequency distributions using the principle of maximum entropy. Advances in Water Resources, Vol. 9, pp. 91106.

    Article  Google Scholar 

  • Singh, V.P., Singh, K. and Rajagopal, A.K., 1985. Application of the principle of maximum entropy (POME) to hydrologic frequency analysis. Completion Report 06, 144 p., Louisiana Water Resources Research Institute, Louisiana State University, Baton Rouge, Baton Rouge, Louisiana.

    Google Scholar 

  • Smith, R.L., 1986. Extreme value theory based on the r largest annual events. Journal of Hydrology, Vol. 86, pp. 27–43.

    Article  Google Scholar 

  • Stol, P.T., 1971. On the decomposition of the extreme value distribution of daily rainfall depths and the derivation of probabilities of compound events. Journal of Hydrology, Vol. 14, pp. 181–196.

    Article  Google Scholar 

  • Wald, A. and Walfowitz, J., 1943. An exact test for randomness in the nonparametric case based on serial correlation. Annals of Mathematical Statistics, Vol. 14, pp. 378–388.

    Article  Google Scholar 

  • Weiss, L.L., 1955. A nomograph based on the theory of extreme values for determining values for various return periods. Monthly Weather Review, Vol. 83, pp. 69–71.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Water Resources Program, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, USA

    Vijay P. Singh

Authors
  1. Vijay P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Singh, V.P. (1998). Extreme Value Type 1 Distribution. In: Entropy-Based Parameter Estimation in Hydrology. Water Science and Technology Library, vol 30. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1431-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1431-0_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5089-2

  • Online ISBN: 978-94-017-1431-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation