Kumaraswamy Normal and Azzalini's skew Normal modeling asymmetry

  • Michelle A Correa
  • Denismar Alves Nogueira
  • Eric Batista Ferreira Professor Adjunto IIIInstituto de Ciências ExatasUniversidade Federal de Alfenas


This paper presents the comparison of two probability distributions with specific parameters for modelling asymmetry. Kum-normal and Azzalini's skew normal distributions were chosen because they turn, in special case, into the normal distribution. The quality of the fit, flexibility and amount of asymmetry parameters were factors used for comparison. Researches state that the Azzalini's skew normal distribution has limitations regarding the flexibility of the tail, presenting certain resistance in modelling asymmetry since, by increasing the absolute value of the asymmetry parameter, it tends to a \emph{half}-normal distribution. The objectives of this study were to implement a kum-normal distribution and, using Monte Carlo simulation to generate data with increasing levels of asymmetry, choose the best fit. The distributions were also compared in modelling a beetle data set (\emph{Tribolium cofusum}), grown at 29°C. For implementation we used the R package \texttt{gamlss}, that allows adjusting of the models, simulating data of generalized distributions and obtaining the Akaike information criterion, Bayesian information criterion and likelihood ratio test, used for comparison. The kum-normal distribution was better adjusted by increasing the level of asymmetry compared to Azzalini's skew normal distribution. For real data the two distributions do not differ significantly, showing equivalent estimation of the degree of asymmetry of these data.

Author Biography

Eric Batista Ferreira, Professor Adjunto IIIInstituto de Ciências ExatasUniversidade Federal de Alfenas

Doutor em Estatística e Experimentação Agropecuária com Pós-doutorado em Estatística Multivariada


AZZALINI, A. A class of distributions which the normal ones. Journal Statistical,v.12, n.2, p.171-178, 1985. CASELLA, G.; BERGER, R. L. Inferência Estatística. São Paulo, 2010, 588p.

CONSTANTINO, R. F.; DESHARNAIS, R. A. Gamma distributions of adult numbers for tribolium populations in the region of their steady states. Journal of Animal Ecology, v.50, p.667-681, 1981.

CORDEIRO, G. M.; CASTRO, M. A new family of generalized distributions. Journal of Statistical Computation & Simulation, v.81, n.7, p.883-898, 2011.

D’AGOSTINO, R. B. Transformation to Normality of the Null. Biometrika, v.57, n.3, p.679-681, 1978.

EUGENE, N.; LEE, C.; FAMOYE, F. Beta-normal distribution and its applications. Communications in Statistics - Theory and Methods, Michigan, USA, v.31, p.497-512, 2002.

FERREIRA, D.F. Estatística Básica. Lavras: UFLA, 2009, 664p.

HOAGLIN, D. C.; PETERS, S. C. Software for Exploring Distributional Shapes. IN: Proceedings of... Computer Science and Statistics: 12th Annual Symposium on the Interface, Ontario, Canada: Iniverty of waterloo, p. 418-443, 1979.

HOAGLIN, D. C. Summarizing shape numerically: The g-and-h Distributions, in Exploring Data, tables, Trends and Shapes. New York: Wiley, p.461-513, 1983.

JONES, M. C. Families of distributions arising from distributions of order statistics. Test, vol 13, n.1, p.1-43, 2004. JONES, M. C. Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, v.6 , p.70-81, 2008.

KUMARASWAMY, P. Generalized probability densy-function for double-bounded random-process. Journal of Hydrology, v.462, p.79-88, 1980.

OLIVEIRA, M. S. Comparações múltiplas Baysianas com erro normal assimétrico. 2009, 154f. Tese (Doutorado em Estatística e Experimentação Agropecuária), Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras, 2009.

R DEVELOPMENT CORE TEAM. R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2011. ISBN 3-900051-07-0, URL http://www.R-project.org/.

SCHWARZ, G. Estimating the dimension of a model. In: STATISTICS, 6, n.2, 1978, Annals... Ann Arbor: Institute of Mathematical Statistics, 1978. pp.461-464.

STASINOPOULOS, D. M.; RIGBY, R. A. Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, v.23, p.1-46, 2007.

TUKEY, J. W. Modern Techniques in Data Analysis. Proceeding of... NSF-Sponsored Regional Research Conference at Southeastern Massachusetts University, North Dartmouth, MA, 1977.