4. Kumaraswamy distribution

Using the same approach as in the Beta distribution, we first study the skewness and kurtosis tendency of KumaraswamyDistribution½α; β� [14], since the latter tested

Figure 10. Contour plot of Kumaraswamy distribution skewness.

Figure 11. Contour plot of Kumaraswamy distribution kurtosis.

Figure 12. Contour plot of Kumaraswamy distribution shape factor.

Figure 13.

Contour plot of Kumaraswamy distribution skewness, kurtosis, and shape factor for given values 5.99, 65.89, and 1.83. The horizontal axis is the α parameter and the vertical axis is the β parameter.

to be a better choice in our experiment and is also the easiest for simulation, Figures 10–12; and then study the SF bound for given skewness.

From these plots, we see an overall rough tendency of the skewness, kurtosis and shape factor. For a given α, the shape factor converges to a finite limit when β ! ∞. For a given skewness or a given kurtosis, there exists a maximum allowable α that is arrived when β ! ∞. In the parameters space of (α,β), for a given α, the kurtosis is increasing with respect to β in the top left portion where the skewness is positive, and decreasing in the right bottom portion where the skewness is negative. And in the parameters space of (α,β), for a given α, the shape factor is decreasing with respect to β in the top left portion where the skewness is positive, and increasing in the right bottom portion where the skewness is negative. But we will see later that the tendencies are more delicate than the monotonicity shown through visual observation.

Combining the tendency of shape factor and the contour plot for given skewness, kurtosis, and shape factor as in Figure 13, we may guess that for a given positive skewness, when α turn to its upper limit and β turn to infinity, the shape
