Abstract
We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical Research Report No. 07/2006. Department of Statistics, University of Brasilia: Brasilia, Brazil, 2006]. Expressions for the moments about origin are derived. Graphical illustrations are also provided. The distributions derived in this paper can be seen as generalizations of the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. Applications with unimodal and bimodal data are given to illustrate the applicability of the results derived in this paper. The applications include the analysis of the following data sets: (a) spending on public education in various countries in 2003; (b) total expenditure on health in 2009 in various countries and (c) waiting time between eruptions of the Old Faithful Geyser in the Yellow Stone National Park, Wyoming, USA. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. The results show that our distributions, in general, fit better the data sets. The general R codes for fitting the distributions introduced in this paper are given in Appendix A.
Highlights
The skew symmetric models have been considered by several researchers
Comparing the proposed distributions in this paper with their corresponding distributions given by Nadarajah and Kotz [6], we can see that: (1) Skew GL-t and Skew Normal-GL distributions have lower values of Akaike criterion (AIC), Bayesian criterion (BIC), AICC, Mean Square Error (MSE), MAD and Max Deviation (MaxD) compared to the Skew Logistic-t and Skew Normal-Logistic distributions, respectively; (2) The Skew t-Logistic gave better accuracy compared to Skew distribution t-GL (Smaller values of MSE, MAD and MaxD), Skew t-GL
Logistic-Normal Skew Logistic-t and Skew Normal-Logistic distributions, respectively; (2) The Skew t-Logistic distribution resulted in better performance when compared to the Skew t-GL distribution (Smaller values of AIC, BIC and AICC), the Skew t-GL distribution obtained better accuracy than Skew t-Logistic distribution (Smaller values of MSE, MAD and MaxD)
Summary
The skew symmetric models have been considered by several researchers. Skew normal distribution is a classical example. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [6], the results show that: (1) our distributions, in general, fit better the data sets; (2) The Skew GL-Normal, Skew GL-t, Skew Normal-GL and Skew t-GL distributions can be used to model symmetrical and asymmetrical unimodal data; (3) The Skew GL-Normal and Skew GL-t distributions can be used to adjust bimodal symmetrical and asymmetrical data, offering good fits, showing a high flexibility which is not common in the literature on probability distributions, which are mostly unimodal This may be very important in practical applications; (4) The distributions are robust to numerical calculations in practical applications. We conclude this introduction section with some results which will be useful in the subsequent sections of this paper
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