Abstract
The generalized gamma (GG) distribution has a density function that can take on many possible forms commonly encountered in hydrologic applications. This fact has led many authors to study the properties of the distribution and to propose various estimation techniques (method of moments, mixed moments, maximum likelihood etc.). We discuss some of the most important properties of this flexible distribution and present a flexible method of parameter estimation, called the “generalized method of moments” (GMM) which combines any three moments of the GG distribution. The main advantage of this general method is that it has many of the previously proposed methods of estimation as special cases. We also give a general formula for the variance of theT-year eventX T obtained by the GMM along with a general formula for the parameter estimates and also for the covariances and correlation coefficients between any pair of such estimates. By applying the GMM and carefully choosing the order of the moments that are used in the estimation one can significantly reduce the variance ofT-year events for the range of return periods that are of interest.
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Ashkar, F.; Bobée, B. 1986: Comment on: Variance of the T-year event in the Log-Pearson type 3 distribution. J. of Hydrology 84, 181–187
Ashkar, F.; Bobée, B. 1988: Confidence intervals for extreme flood events under a Pearson type 3 or log Pearson type 3 distribution. Water Resour Bulletin (accepted for publication)
Benson, M.A. 1968: Uniform flood frequency estimating methods for federal agencies. Water Resour. Res. 4, 891–908
Bobée, B.; Boucher, P. 1981: Calcul de la variance d'un évènement de période de retour T: cas des lois log-Pearson type 3 et log-gamma ajustés par la méthode des moments sur la série des valeurs observées. INRS-Eau, Rapp. Sci. 135, 17 pp
Hager, H.W.; Bain, L.J. 1970: Inferential procedures for the generalized gamma distribution. J. Amer. Statist. Assoc. 65 1601–1609
Harter, H.L. 1969: A new table of percentage points of the Pearson type 3 distribution. Technometrics 11, 177–187
Hoshi, K.; Burges, S.J. 1981: Approximate estimation of the derivative of a standard gamma quantile for use in confidence interval estimates. J. of Hydrology 53, 317–325
Hoshi, K.; Yamaoka, I. 1980: The generalized gamma probability distribution and its application in hydrology. Hydraulics papers 7. The research laboratory of civil and environmental engineering, Hokkaido University, Japan, 256 pp
Kendall, M.J.; Stuart, A.; Ord, J.K. 1987: Kendall's advanced theory of statistics. Vol. 1. 5th ed., Oxford University Press, New York, 604 pp
Kite, G.W. 1977: Frequency and risk analysis in hydrology. Water Resour. Publications. Colorado: Fort Collins, 224 pp
Kritsky, S.N.; Menkel, M.F. 1969: On principles of estimation methods of maximum discharge. Floods and their computation. A.I.H.S. 84
Leroux, D; Bobée, B.; Ashkar, F. 1987: Diagrammes des relations entre certaines fonctions de moments pour quelques lois fréquemment utilisées en hydrologie. INRS-Eau, Rapp. Sci. 237
Nicholson, D.L. 1975: Application, of parameter estimation and hypothesis test for a generalized gamma distribution. Proc. IEEE conf. decis. control incl. symp. adapt. processes. Houston, Tex., Dec 10–12, 321–326
Paradis, M.; Bobée, B. 1983: La distribution Gamma Généralisée et son application en hydrologie. INRS-Eau, Rapp. Sci. 156, 52 pp
Parr, V.B.; Webster, J.T. 1965: A method for discriminating between failure density functions used in reliability predictions. Technometrics, 7, 1–10
Phien, H.N.; Nguyen, V.T.V.; Jung-Hua, K. 1987: Estimating the parameters of the generalized gamma distribution by mixed moments. In: V.P. Singh (ed.), Hydrologic Frequency Modeling. D. Reidel Publ. Co., Dordrecht, Holland
Rao, D.V. 1980: Log Pearson type 3 distribution: Method of mixed moments. J. Hydraul. Div. Am. Soc. Civ. Eng. 106, 999–1019
Sangal, B.P.; Biswas, A.K. 1970: The 3-parameter log normal distribution and its applications in hydrology. Water Resour. Res. 2, 505–515
Stacy, E.W. 1962: A generalization of the gamma distribution. Ann. of Math. Stat. 33, 1187–1192
Stacy, E.W.; Mihram, G.A. 1965: Parameter estimation for a generalized gamma distribution. Technometrics 7, 349–358
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Ashkar, F., Bobée, B., Leroux, D. et al. The generalized method of moments as applied to the generalized gamma distribution. Stochastic Hydrol Hydraul 2, 161–174 (1988). https://doi.org/10.1007/BF01550839
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DOI: https://doi.org/10.1007/BF01550839