A Modified Hybrid Gamma and Generalized Pareto Distribution for Precipitation Data

  • Yongku Kim
  • Hyeongang Kim
  • GyuWon Lee
  • Ki-Hong MinEmail author
Original Article


This study introduces a modified hybrid gamma and generalized Pareto distribution. Prior to this, we define a general spliced distribution and its corresponding gamma distribution, which is part of the head, and a generalized Pareto (GP) distribution, which is part of the tail. We then examine the threshold conditions for the modified hybrid gamma and GP distribution and defined probability density function. Also, we derive the negative log-likelihood function of the modified hybrid gamma and GP distribution and estimate approximate maximum likelihood estimates using the differential evolution algorithm for each simulation to minimize it. Moreover, by presenting the mean square error for each sample size, the model is evaluated according to the size of the sample. Finally, we use daily observed summer precipitation for Seoul, Korea, from 1961 to 2011, which includes 4692 data sets. We use 2051 data sets corresponding to wet conditions. As a result, the estimated threshold of the modified hybrid gamma and GP distribution is 0.1455. After deriving Fisher information through the Hessian matrix, we also present the standard error of the maximum likelihood estimator.


Gamma distribution Generalized Pareto distribution Hybrid distribution Precipitation 



This subject is supported by Korea Ministry of Environment (MOE) as “Water Management Research Program” and by “Development of Nowcasting Applications Algorithms” project, funded by ETRI, which is a subproject of “Development of Geostationary Meteorological Satellite Ground Segment (NMSC-2018-01)” program funded by NMSC (National Meteorological Satellite Center) of KMA (Korea Meteorological Administration).


  1. Bakar, S., Hamzah, N.A., Maghsoudi, M., Nadarajah, S.: Modeling loss data using composite models. Insur. Math. and Econ. 61, 146–154 (2015)CrossRefGoogle Scholar
  2. Baxevani, A., Lennartsson, L.: A spatiotemporal precipitation generator based on a censored latent Gaussian field. Water Resour. Res. 51, 4338–4358 (2015)CrossRefGoogle Scholar
  3. Benestad, R. E., I. Hanssen-Bauer, and D. Chen, 2008: Empirical Statistical Downscaling. World Scientific Publishing Company, 228 pp.Google Scholar
  4. Buishand, T.A.: Some remarks on the use of daily rainfall models. J. Hydrol. 47, 235–249 (1978)Google Scholar
  5. Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Amer. Stat. Assoc. 74, 829–836 (1979)CrossRefGoogle Scholar
  6. Coe, R., Stren, R.D.: Fitting models to daily rainfall data. J. Appl. Meteorol. 21, 1024–1031 (1982)CrossRefGoogle Scholar
  7. Deguenon, J., Barbulescu, A., Sarr, M.: GPD models for extreme rainfall in Dobrudja. Comput. Eng. Sys. Appl. 2, 131–136 (2009)Google Scholar
  8. Furrer, E.M., Katz, R.W.: Generalized linear modeling approach to stochastic weather generators. Clim. Res. 34, 129–144 (2007)CrossRefGoogle Scholar
  9. Hanum, H., A. Hamim, A., A. Djuraidah, and W. Mangku, 2015: Modeling extreme rainfall with gamma-Pareto distribution. Appl. Math. Sci., 9, 6029–6039Google Scholar
  10. Hastie, T.J., Tibshirani, R.J.: Generalized additive models, p. 352. Chapman and Hall (1990)Google Scholar
  11. Katz, R.W., Parlange, M.B.: Overdispersion phenomenon in stochastic modeling of precipitation. J. Clim. 11, 591–601 (1998)CrossRefGoogle Scholar
  12. Kim, Y., Lee, G.W.: Stochastic precipitation generator with hidden state covariates. Asia-Pac. J. Atmos. Sci. 53(3), 353–359 (2017)CrossRefGoogle Scholar
  13. Kim, Y., Katz, R.W., Rajagopalan, B., Podest, G.P., Furrer, E.M.: Reducing overdispersion in stochastic weather generators using a generalized linear modeling approach. Clim. Res. 53, 13–24 (2012)CrossRefGoogle Scholar
  14. Klugman, S.A., Panjer, H.H., Willmot, G.E.: Loss models: From data to decisions, 2nd edn. Wiley, New York (2004) 720ppGoogle Scholar
  15. Li, C., Singh, V.P., Mishra, A.K.: Simulation of the entire range of daily precipitation using a hybrid probability distribution. Water Resour. Res. 48, W03521 (2012)Google Scholar
  16. Nadarajah, S., Bakar, S.: New composite models for the Danish reinsurance data. Scand. Actuar. J. 2014, 180–187 (2014)CrossRefGoogle Scholar
  17. Scollnik, D.P.M.: On composite lognormal-Pareto models. Scand. Actuar. J. 2007, 20–33 (2007)CrossRefGoogle Scholar
  18. Wilks, D.S., Wilby, R.L.: The weather generator game: a review of stochastic weather models. Prog. Phys. Geogr. 23, 329–357 (1999)CrossRefGoogle Scholar

Copyright information

© Korean Meteorological Society and Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of StatisticsKyungpook National UniversityDaeguSouth Korea
  2. 2.Department of Astronomy and Atmospheric SciencesKyungpook National UniversityDaeguSouth Korea
  3. 3.Center for Atmospheric REmote Sensing (CARE)Kyungpook National UniversityDaeguSouth Korea
  4. 4.Department of Astronomy and Atmospheric Sciences, College of Natural ScienceKyungpook National UniversityDaeguSouth Korea

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