Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T18:26:27.644Z Has data issue: false hasContentIssue false

1 - Introduction

Published online by Cambridge University Press:  13 April 2022

Vijay P. Singh  and
Lan Zhang
Vijay P. Singh
Affiliation:
Texas A & M University
Lan Zhang
Affiliation:
University of Akron, Ohio
Get access

Summary

Several generalized frequency distributions have been employed in environmental and water engineering over the years. These distributions are quite versatile and can apply to frequency analysis of a wide variety of random variables, such as flood peaks, volume, duration, and inter-arrival time; extreme rainfall amount, duration, spatial coverage, and inter-arrival time; drought duration, severity, spatial extent, and inter-arrival time; wind speed, duration, direction, and spatial coverage; water quality parameters; and sediment concentration, discharge, and yield. However, because of their relatively complex form, these distributions have not become as popular as the simpler distributions. These distributions have at least three but usually more parameters, which have been estimated using the methods of moments, maximum likelihood, probability weighted moments, and L-moments. In some cases, entropy theory has been used to estimate parameters. This chapter provides a snapshot of the generalized distributions that will be discussed in this book. Moreover, a short discussion of the methods of parameter estimation, goodness-of-fit statistics, and confidence intervals is provided.

Keywords

distributionsmethod of momentmaximum likelihoodprobability weighted momentL-moment
Type
Chapter
Information
Generalized Frequency Distributions for Environmental and Water Engineering , pp. 1 - 49
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T.W. and Darling, D.A. (1952). Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Annals of Mathematical Statistics, Vol. 23, pp. 193–212.CrossRefGoogle Scholar
Ashkar, F., Bobée, B., Leroux, D., and Morisette, D. (1988). The generalized method of moments as applied to the generalized gamma distribution. Stochastic Hydrology and Hydraulics, Vol. 2, pp. 161–174.Google Scholar
Barlow, R.E. and Proschan, F. (1965). Mathematical Theory of Reliability. John Wiley & Sons, Inc, New York.Google Scholar
Bobée, B., Perreault, L., and Ashkar, F. (1993). Two kinds of moment diagrams and their applications in hydrology. Stochastic Hydrology and Hydraulics, Vol. 7, pp. 41–65.Google Scholar
Burr, I.W. (1942). Cumulative frequency functions. The Annals of Mathematical Statistics, Vol. 13, No. 2, pp. 215–232.Google Scholar
Clarke, R.T. (1996). Residual maximum likelihood (REML) methods for analyzing hydrological data series. Journal of Hydrology, Vol. 182, pp. 277–295.Google Scholar
Cramér, H. (1928). On the composition of elementary errors. Scandinavian Actuarial Journal, Vol. 1, pp. 13–74. doi: 10.1080/03461238.1928.10416862.Google Scholar
Cunnane, C. (1989). Statistical Distribution for Flood Frequency Analysis. WMO Operational Hydrology Report No. 33, WMO-No. 718, Geneva, Switzerland.Google Scholar
De Haan, D.B. (1867). Nouvelles tables d’Integrales definies. G. E. Stechert, New York.Google Scholar
Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R. (1979). Probability-weighted moments: Definition and relation to parameters of several distributions expressible in inverse form. Water Resources research, Vol. 15, pp. 1049–1054.Google Scholar
Haktamir, T. (1996). Probability-weighted moments without plotting position formula. Journal of Hydrologic Engineering, Vol. 1, No. 2, pp. 89–91.Google Scholar
Hosking, J.R.M. (1985). A correction for the bias of maximum likelihood estimators of Gumbel parameters – Comment. Journal of Hydrology, Vol. 78, pp. 393–396.Google Scholar
Hosking, J.R.M. (1986). The Theory of Probability-Weighted Moments. Technical Report RC 12210, Mathematics, 160 pp., IBM Thomas Watson Research Center, Yorktown Heights, New York.Google Scholar
Hosking, J.R.M. (1990). L-moments: Analysis and estimation of distribution using linear combination of order statistics. Journal of Royal Statistical Society, Series B, Vol. 52, No. 1, pp. 105–124.Google Scholar
Jaynes, E.T. (1957). Information theory and statistical mechanics, I. Physical Review, Vol. 106, pp. 620–630.Google Scholar
Koch, S.P. (1991). Bias error in maximum likelihood estimation. Journal of Hydrology, Vol. 122, pp. 289–300.CrossRefGoogle Scholar
Kolmogorov, A. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari, Vol. 4, pp. 83–91.Google Scholar
Landwehr, J.M., Matalas, N.C., and Wallis, J.R. (1979a). Probability-weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resources Research, Vol. 15, pp. 1055–1064.Google Scholar
Landwehr, J.M., Matalas, N.C., and Wallis, J.R. (1979b). Estimation of parameters and quantiles of Wakeby distribution. Water Resources Research, Vol. 15, pp. 1361–1379.Google Scholar
Liao, M. and Shimokawa, T. (1999). A new goodness-of-fit for type I extreme value and 2-parameter Weibull distributions with estimated parameters. Optimization, Vol. 64, No. 1, pp. 23–48.Google Scholar
von Mises, R.E. (1928). Wahrscheinlichkeit, Statistik und wahreit. Julius Springer,Vienna.Google Scholar
Shannon, C.E. (1948). The mathematical theory of communication, I and II. Bell System Technical Journal, Vol. 27, pp. 379–423.Google Scholar
Singh, V.P. (1988). Hydrologic Systems, Vol. 1: Rainfall-Runoff Modeling. Prentice Hall, Engelwood Cliffs, NJ.Google Scholar
Singh, V.P. (1998). Entropy-Based Parameter Estimation in Hydrology. Kluwer Academic Publishers (now Springer), Dordrecht, the Netherlands.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
Smirnov, N. (1948). Table for estimating the goodness-of-fit of empirical distributions. Annals of Mathematical Statistics, Vol. 19, pp. 279–281. doi: 10.1214/aoms/1177730256.Google Scholar
Sorooshian, S., Gupta, V.K., and Fulton, J.L. (1983). Evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall-runoff models: Influence of calibration data variability and length on model credibility. Water Resources Research, Vol. 19, No. 1, pp. 251–259.Google Scholar
Stock, J.H. and Watson, M.W. (1989). Interpreting the evidence on money-income casualty. Journal of Econometrics, Vol. 40, pp. 161–181.Google Scholar
Vogel, R.M. and Fennessey, N.M. (1993). L-moment diagrams should replace product moment diagrams. Water Resources research, Vol. 29, No. 6, pp. 1745–1752.Google Scholar
Wang, Q.J. (1997). LH moments for statistical analysis of extreme events. Water Resources Research, Vol. 33, No. 12, pp. 2841–2848.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Introduction
  • Vijay P. Singh, Texas A & M University, Lan Zhang, University of Akron, Ohio
  • Book: Generalized Frequency Distributions for Environmental and Water Engineering
  • Online publication: 13 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781009025317.002
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Introduction
  • Vijay P. Singh, Texas A & M University, Lan Zhang, University of Akron, Ohio
  • Book: Generalized Frequency Distributions for Environmental and Water Engineering
  • Online publication: 13 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781009025317.002
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Introduction
  • Vijay P. Singh, Texas A & M University, Lan Zhang, University of Akron, Ohio
  • Book: Generalized Frequency Distributions for Environmental and Water Engineering
  • Online publication: 13 April 2022
  • Chapter DOI: https://doi.org/10.1017/9781009025317.002
Available formats
×