Beta-\(\kappa \) distribution and its application to hydrologic events. (English) Zbl 1221.62027

Summary: The beta-\(\kappa \) distribution is a distinct case of the generalized beta distribution of the second kind. In previous studies, beta-\(p\) and beta-\(\kappa \) distributions have played important roles in representing extreme events, and thus, the present paper uses the beta-\(\kappa \) distribution. Further, this paper uses the method of moments and the method of L-moments to estimate the parameters from the beta-\(\kappa \) distribution, and to demonstrate the performance of the proposed model, the paper presents a simulation study using three estimation methods (including the maximum likelihood estimation method) and beta-\(\kappa \) and non beta-\(\kappa \) samples. In addition, this paper evaluates the performance of the beta-\(\kappa \) distribution by employing two widely used extreme value distributions (i.e., the GEV and Gumbel distributions) and two sets of actual data on extreme events.


62E15 Exact distribution theory in statistics
62F10 Point estimation
62G32 Statistics of extreme values; tail inference
86A05 Hydrology, hydrography, oceanography
65C60 Computational problems in statistics (MSC2010)
86A32 Geostatistics
Full Text: DOI


[1] Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, Washington, DC · Zbl 0171.38503
[2] Anderson TW, Darling DA (1954) A test of goodness of fit. J Am Stat Assoc 49:765-769 · Zbl 0059.13302 · doi:10.2307/2281537
[3] Bernardara P, Schertzer D, Sauquet E (2008) The flood probability distribution tail: how heavy is it? Stoch Env Res Risk Assess 22:107-122 · Zbl 1169.62395 · doi:10.1007/s00477-006-0101-2
[4] Castillo E, Hadi AS, Balakrishnan N, Sarabia JM (2005) Extreme value and related models with applications in engineering and science. Wiley-Interscience, Hoboken · Zbl 1072.62045
[5] Chebana F, El Adlouni S, Bobee B (2008) Method of moments of the Halphen distribution parameters. Stoch Env Res Risk Assess 22:749-757 · Zbl 1409.62053 · doi:10.1007/s00477-007-0184-4
[6] Chebana F, El Adlouni S, Bobee B (2010) Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events. Stoch Env Res Risk Assess 24:359-376 · Zbl 1420.62225 · doi:10.1007/s00477-009-0325-z
[7] Coles S (2001) An introduction to statistical modeling of extreme values. Springer, London · Zbl 0980.62043
[8] Coles S, Dixon M (1999) Likelihood-based inference for extreme value models. Extremes 2:5-23 · Zbl 0938.62013 · doi:10.1023/A:1009905222644
[9] El Adlouni S, Bobee B, Ouarda TBMJ (2008) On the tails of extreme event distributions in hydrology. J Hydrol 355:1633 · doi:10.1016/j.jhydrol.2008.02.011
[10] Greenwood JA, Landwehr NC, Matalas, Wallis JR (1979) Probability weighted moments: definition and relation to parameters of distribution expressible in inverse form. Water Resour Res 15:1049-1054
[11] Hosking JRM (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc B 52:105-124 · Zbl 0703.62018
[12] Hosking JRM (2000) LMOMENTS: Fortran routines for use with the method of L-moments, version 3.03. http://www.research.ibm.com/people/h/hosking/lmoments.html
[13] Hosking JRM, Wallis JR (1995) A comparison of unbiased and plotting-position estimators of L-moments. Water Resour Res 31:2019-2025 · doi:10.1029/95WR01230
[14] Hosking JRM, Wallis JR (1997) Regional frequency analysis: an approach based on L-moments. Cambridge University Press, Cambridge · doi:10.1017/CBO9780511529443
[15] Hosking JRM, Wallis JR, Wood EF (1985) Estimation of the generalized extreme value distribution by the method of probability-weighted moments. Technometrics 27:251-261 · doi:10.2307/1269706
[16] Houghton JC (1978) Birth of parent: the Wakeby distribution for modeling flood flows. Water Resour Res 14(6):1111-1115 · doi:10.1029/WR014i006p01111
[17] Landwehr JM, Matalas NC (1980) Quantile estimation with more or less floodlike distribution. Water Resour Res 16(3):547-555 · doi:10.1029/WR016i003p00547
[18] Martin ES, Stedinger JR (2000) Generalized maximum likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resour Res 36:737-744 · doi:10.1029/1999WR900330
[19] Mason SJ, Waylen PR, Mimmack GM et al (1999) Changes in extreme rainfall events in South Africa. Clim Chang 41:249-257 · doi:10.1023/A:1005450924499
[20] Meshgi A, Kahlili D (2009) Comprehensive evaluation of regional flood frequency analysis by L- and LH-moments. I. A re-visit to regional homogeneity. Stoch Env Res Risk Assess 23:119-135 · Zbl 1409.62101 · doi:10.1007/s00477-007-0201-7
[21] Mielke PW (1973) Another family of distributions for describing and analyzing precipitation data. J Appl Meteorol 12:275-280 · doi:10.1175/1520-0450(1973)012<0275:AFODFD>2.0.CO;2
[22] Mielke PW, Johnson ES (1973) Three parameter Kappa distribution maximum likelihood estimations and likelihood ratio tests. Mon Water Rev 101:701-707 · doi:10.1175/1520-0493(1973)101<0701:TKDMLE>2.3.CO;2
[23] Mielke PW, Johnson ES (1974) Some generalized beta distributions of the second kind having desirable application features in hydrology and meteorology. Water Resour Res 10:223-226 · doi:10.1029/WR010i002p00223
[24] Oztekin T (2007) Wakeby distribution for representing annual extreme and partial duration rainfall series. Meteorol Appl 14:381-387 · doi:10.1002/met.37
[25] Park JS (2005) A simulation-based hyperparameter selection for quantile estimation of the generalized extreme value distribution. Math Comput Simul 70:227-234 · Zbl 1077.62040 · doi:10.1016/j.matcom.2005.09.003
[26] Park JS, Jung HS, Kim RS, Oh JH (2001) Modelling summer extreme rainfall over the Korean peninsula using Wakeby distribution. Int J Climatol 21:1371-1384
[27] Park JS, Seo SC, Kim TY (2009) A kappa distribution with a hydrological application. Stoch Env Res Risk Assess 23:579-586 · Zbl 1411.86010 · doi:10.1007/s00477-008-0243-5
[28] Ross SM (1997) Simulation, 2nd edn. Academic Press, San Diego · Zbl 0898.65004
[29] Serinaldi F (2009) Assessing the applicability of fractional order statistics for computing confidence intervals for extreme quantiles. J Hydrol 376(3-4):528-541 · doi:10.1016/j.jhydrol.2009.07.065
[30] Stream flow data of Colorado from U.S. Geological Survey (USGS). http://nwis.waterdata.usgs.gov/co/nwis/peak?site_no=08230500&agency_cd=USGS&format=html. Accessed June 2010
[31] Wang QJ (1997) LH moments of statistical analysis of extreme events. Water Resour Res 33:2841-2848 · doi:10.1029/97WR02134
[32] Wilks DS (1993) Comparison of three-parameter probability distributions for representing annual extreme and partial duration precipitation series. Water Resour Res 29:3543-3549 · doi:10.1029/93WR01710
[33] Wilks DS, Mckay M (1996) Extreme-value statistics for snowpack water equivalent in northeastern United States using the cooperative observar network. J Appl Meteorol 35:706-713 · doi:10.1175/1520-0450(1996)035<0706:EVSFSW>2.0.CO;2
[34] Wolfram S (1991) Mathematica, a system for doing mathematics by computer. Addison-Wesley, Reading · Zbl 0671.65002
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