Kumaraswamy Normal and Azzalini's skew Normal modeling asymmetry

Abstract

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.