LH-Moments of Some Distributions Useful in Hydrology

Murshed, Md. Sharwar;Park, Byung-Jun;Jeong, Bo-Yoon;Park, Jeong-Soo

  • 발행 : 2009.07.31


It is already known from the previous study that flood seems to have heavier tail. Therefore, to make prediction of future extreme label, some agreement of tail behavior of extreme data is highly required. The LH-moments estimation method, the generalized form of L-moments is an useful method of characterizing the upper part of the distribution. LH-moments are based on linear combination of higher order statistics. In this study, we have formulated LH-moments of five distributions useful in hydrology such as, two types of three parameter kappa distributions, beta-${\kappa}$ distribution, beta-p distribution and a generalized Gumbel distribution. Using LH-moments reduces the undue influences that small sample may have on the estimation of large return period events.


Beta-${\kappa}$ distribution;beta-p distribution;generalized Gumbel distribution;L-moment;probability weighted moment;three parameter kappa distribution


  1. Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London
  2. Gradshteyn, I. S. and Ryzhik, M. (1980). Table of Integrals, Series and Products, 4th ed., Academic Press, New York
  3. Greenwood, J. A., Landwehr, J. M., Matalas, N. C. and Wallis, J. R. (1979). Probability weighted moments: Definition and relation to parameters of distribution expressible in inverse form, Water Resources Research, 15, 1049-1054
  4. Hosking, J. R. M. (1990). L-moments: Analysis and estimation of distributions using linear combina-tions of order statistics, Journal of Royal Statistical Society, Series B, 52, 105-124
  5. Hosking, J. R. M. (1994). The four-parameter Kappa distribution, IBM Journal of Research and Development, 38, 251-258
  6. Hosking, J. R. M. and Wallis J. R. (1997). Regional Frequency Analysis: An Approach Based on L-moments, Cambridge University Press, Cambridge, UK
  7. Jeong, B. Y. (2009). On the Properties of a Generalized Gumbel Distribution and r-kappa Distribu-tion, Ph.D. Thesis, Chonnam National University, Gwangju
  8. Johnson, N. L. and Kotz, S. (1970). Continuous Univariate Distributions-1, Wiley Interscience, New York
  9. Lee, S. H. and Maeng, S. J. (2003). Comparison and analysis of design floods by the change in the order of LH-moment methods, Irrigation and Drainage, 52, 231-245
  10. Mason, S. J., Waylen, P. R., Mimmack, G. M., Rajaratnam, B. and Harrison, J. M. (1999). Changes in extreme rainfall events in South Africa, Climatic Change, 41, 249-257
  11. Meshgi, A. and Khalili, D. (2009). Comprehensive evaluation of regional flood frequency analysis by L- and LH-moments. I. A re-visit to regional homogeneity, Stochastic Environmental Research and Risk Assessment, 23, 119-135
  12. Mielke, P. W. (1973). Another family of distributions for describing and analyzing precipitation data, Journal of Applied Meteorology, 12, 275-280<0275:AFODFD>2.0.CO;2
  13. Mielke, P. W. and Johnson, E. S. (1973). Three-parameter kappa distribution maximum likelihood estimates and likelihood ratio tests, Monthly Weather Review, 101, 701-711<0701:TKDMLE>2.3.CO;2
  14. Mielke, P. W. and Johnson, E. S. (1974). Some generalized beta distributions of the second kind having desirable application features in hydrology and meteorology, Water Resources Research, 10, 223-226
  15. Oh, M., Kim, S., Park, J. S. and Son, Y. S. (2007). Bayesian estimation of the two-parameter kappa distribution, Communications of the Korean Statistical Society, 14, 355-363
  16. Oztekin, T. (2007). Wakeby distribution for representing annual extreme and partial duration rainfall series, Meteorological Applications, 14, 381-387
  17. Park, J. S., Jung, H. S., Kim, R. S. and Oh, J. H. (2001). Modelling summer extreme rainfall over the Korean peninsula using Wakeby distribution, International Journal of Climatology. 21, 1371-1384
  18. Park, J. S. and Jung, H. S. (2002). Modelling Korean extreme rainfall using a Kappa distribution and maximum likelihood estimate, Theoretical and Applied Climatology, 72, 55-64
  19. Park, J. S. and Kim, T. Y. (2007). Fisher information matrix for a four-parameter Kappa distribution, Statistics & Probability Letters, 77, 1459-1466
  20. Park, J. S., Seo, S. C. and Kim, T. Y. (2009). A kappa distribution with a hydrological application, Stochastic Environmental Research and Risk Assessment, 23, 579-586
  21. Wang, Q. J. (1997). LH moments of statistical analysis of extreme events, Water Resources Research, 33, 2841-2848
  22. Wilks, D. S. (1993). Comparison of three-parameter probability distributions for representing annual extreme and partial duration precipitation series, Water Resources Research, 29, 3543-3549