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.

MSC:

 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

Software:

ismev; lmoments; LMOMENTS; Mathematica
Full Text:

References:

 [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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