Approximate Confidence Intervals for Quantiles of Gamma and Generalized Gamma Distributions
Publication: Journal of Hydrologic Engineering
Volume 3, Issue 1
Abstract
In hydraulic design, one often needs to estimate flood quantiles for use as design values. It is important to assess the estimation error by constructing confidence intervals (CIs) for these quantiles. Fitting probability distributions to hydrologic data is used widely for estimating quantiles of hydrological variables. The two-parameter gamma (G2) is among the distributions commonly used, but the three-parameter generalized gamma (GG3) (also known as Kritsky-Menkel distribution) is an alternative when more shape flexibility is needed to fit the data. We use an approximate method to construct CIs for the quantiles of the G2 and GG3 distributions. This method is shown to be useful for hydrological applications where the data record is short.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Abramowitz, M., and Stegun, I. (eds.) (1965). Handbook of mathematical functions. Dover Publications, New York, N.Y.
2.
Ashkar, F., and Bobée, B.(1988). “Confidence intervals for flood events under a Pearson 3 or log-Pearson 3 distribution.”Water Resour. Bull., 24(3), 639–650.
3.
Ashkar, F., and Ouarda, T. B. M. J. (1994). “Tolerance limits for the gamma and generalized gamma distributions.”Scientific Rep. No. STAT-12, Dept. of Math., Univ. of Moncton, Moncton, N.B., Canada.
4.
Ashkar, F., Bobée, B., Leroux, D., and Morisette, D.(1988). “The generalized method of moments applied to the generalized gamma distribution.”Stochastic Hydrol. and Hydr., 2, 161–174.
5.
Bain, L. J., and Engelhardt, M.(1981). “Simple approximate distributional results for confidence and tolerance limits for the Weibull distribution based on maximum likelihood estimators.”Technometrics, 23(1), 15–20.
6.
Bain, L. J., and Engelhardt, M. (1992). Introduction to probability and mathematical statistics, 2nd Ed., PWS-KENT Publishing Co., Boston, Mass.
7.
Bain, L. J., Engelhardt, M., and Shiue, W. K. (1984). “Approximate tolerance limits and confidence limits on reliability for the gamma distribution.”IEEE Trans. on Reliability R-33, (2), 184–187.
8.
Bobée, B., and Ashkar, F. (1991). The gamma family and derived distributions applied in hydrology. Water Resources Publications, Littleton, Colo.
9.
Chowdhury, J. U., and Stedinger, J. R.(1991). “Confidence interval for design floods with estimated skew coefficient.”J. Hydr. Engrg., ASCE, 117(7), 811–831.
10.
D'Agostino, R. B., and Stephens, M. A. (1986). Goodness-of-fit techniques. Marcel Dekker, New York, N.Y.
11.
Hahn, G. J., and Meeker, W. Q. (1991). Statistical intervals/a guide for practitioners. J. Wiley & Sons, Inc., New York, N.Y.
12.
Johnson, N. L., and Welch, B. L.(1940). “Applications of the non-central T-distribution.”Biometrika, 31, 362–389.
13.
Lawless, J. F. (1982). Statistical models and methods for lifetime data. John Wiley & Sons, Inc., New York, N.Y.
14.
Mann, N. R., and Fertig, K. W.(1977). “Efficient unbiased quantile estimators for moderate-size complete samples from extreme value and Weibull distributions: confidence bounds and tolerance and prediction intervals.”Technometrics, 19, 87–93.
15.
Odeh, R. E., and Owen, D. B. (1980). Tables for normal tolerance limits, sampling plans and screening. Marcel Dekker, New York, N.Y.
16.
Stedinger, J. R.(1983). “Confidence intervals for design events.”J. Hydr. Engrg., ASCE, 109(1), 13–27.
17.
Wilson, E. B., and Hilferty, M. M.(1931). “The distribution of chi-square.”Proc., Nat. Acad. of Sci., 17(12), 684–688.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 3 • Issue 1 • January 1998
Pages: 43 - 51
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Jan 1, 1998
Published in print: Jan 1998
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.