A proposed reparametrization of gamma distribution for the analysis of data of rainfall-runoff driven pollution

  • Bernardo Lagos Universidad de Concepción.
  • G. Ferreira Universidad de Concepción.
  • M. Valenzuela Universidad de Concepción.
Palabras clave: Gamma distribution, Maximum likelihood estimators, Moment estimators, Probability weighted moment estimators, Weighted least squares estimators, Water pollution and watershed.


A generalized gamma (GG) distribution of four parameters was first introduced by Amoroso 1925, and since then, different distributions emerged as subclasses of this model. This model is commonly used to model lifetime data or data with a right skewed unimodal density function. In this article, we use a reparameterization of the GG distribution that is compared with other usual two-parameter distributions, Weibull, generalized exponential (Gupta and Kundu 1999), and gamma, using a real data set with a high coefficient of asymmetry and kurtosis (Valenzuela M. 2009). Akaike's information criterion and Bayesian information criterion indicates that our reparametrization of the gamma distribution is better. Besides a Monte Carlo simulation study, shows the behavior of five estimation methods: least squared, weighted least squared, moments, probability weighted moments and maximum likelihood methods.


[1] Amoroso, L. : Ricerche intorno alla curva dei redditi (in Italian). Ann. Math. Pura Appl. 4 (21), pp. 123-159, (1925).

[2] Anselin, L., Spatial Econometrics: Methods and Models. Kluwer Academic Publisher, The Nederlands, (1988).

[3] Clesceri, L. S., G. A., and Eaton, A., Standard methods for the examination of water and wastewater, 20th Edition, Washington DC: American Public Health Association (APHA), (1998).

[4] Cox, C., Chu, H., Schneider, M.F. and Muñoz, A. Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine, 26, pp. 4352-4374, (2007).

[5] Cox, D.R. and Hinkley, D.V. (1974) Theoretical Statistics, London: Chapman and Hall.

[6] Crooks G. E (2010) The Amoroso Distribution, Technical Notes, Physical Biosciences Division, Lawrence Berkeley National Lab, Berkeley.

[7] Gradshteyn, I. S. and Ryzhik, I. M. (2007) Table of Integrals, Series and Products, Seventh Edition, Academic Press, San Diego, U.S.A., (2007).

[8] Greenwood, J. A. Landwehr, J. M. Matalas, N. C. and Wallis J. R. Probability Weighted Moments: Definition and Relation to Parametes of several Distributions expressible in inverse Form. Water Resources Research, 15, pp. 1049-1054, (1979).

[9] Gupta, R. D. and Kundu, D. Generalized Exponential Distributions, Australian and New Zealand Journal of Statistics, 41 (2), pp. 173-188, (1999).

[10] Gupta, R. D. and Kundu, D. Exponentiated Exponential Family: an alternative to Weibull and Gamma Distribution. Biometrical Journal, 43, pp. 117-130, (2001).

[11] Johnson, N. L., Kotz, S. and Balakrishnan, N. Continuous Univariate Distributions, Vol. 1, Wiley, New York, (1994).

[12] Konn, H. and Lomdahl, P. S. (2004) Stochastic Processes Having Fractional Order Nonlinearity Associated with Hyper Gamma Distribution, Journal of the Physical Society of Japan, 73(3), pp. 573-579, (2004).

[13] Lawless, J. F. Inference in the generalized gamma and log gamma distributions. Technometrics, 22, pp. 409-419, (1980).

[14] Mooley, D. A., and Mohile, C. M., Some aspects of rainfall associated with cyclonic storms of the Bay of Bengal, International Journal of Climatology, 6, 149-160, (1986).

[15] Mooley, D., and Appa Rao, G., Distribution function for seasonal and annual rainfall over India, Monthly Weather Review, 99, 796-799, (1971).

[16] Murray, C., Sampling and data analysis for environmental microbiology, in Manual of Environmental Microbiology. 2nd ed, eds. K. M. Hurst, Crawford, and S. L.D., Washington D. C., USA: ASM Press, (2002).

[17] Noufaily, A. and Jones, M. C. On Maximization of the Likelihood for the Generalized Gamma Distribution,in Technical Report in Statistics, Department of Mathematics and Statistics, The Open University, (2009).

[18] Stacy, E. W. A generalization of the gamma distribution, Ann. Math. Stat. 33, pp. 1187-1192, (1962).

[19] Stacy, E., and Mihram, G. (1965), Parameter estimation for a generalized gamma distribution, Technometrics, 7, 349-358.

[20] Suzuki, E. Hyper gamma distribution and its fitting to rainfall data. Pap. Meteor. Geophys. 15, pp. 31-51, (1964).

[21] Swain, J. , Venkatraman, S. and Wilson, J. Least squares estimation of distribution function in Johnson’s translation system, Journal of Statistical Computation and Simulation 29, pp. 271-297, (1988).

[22] Valenzuela Mariella., Lagos B., Claret M., Mondaca M., Parra O. Ocurrence of faecal contamination in groundwater at a small rural watershed. Chilean Journal of Agricultural Research, 69 (2), pp. 235-243, (2009).
Cómo citar
Lagos, B., Ferreira, G., & Valenzuela, M. (2011). A proposed reparametrization of gamma distribution for the analysis of data of rainfall-runoff driven pollution. Proyecciones. Revista De Matemática, 30(3), 415-439. https://doi.org/10.4067/S0716-09172011000300009