Shape Factor Asymptotic Analysis I

The shape factor defined as kurtosis divided by skewness squared K/S^2 is characterized as the only choice among all factors K/〖|S|〗^α ,α>0 which is greater than or equal to 1 for all probability distributions. For a specific distribution family, there may exists α>2 such that min⁡〖K/〖|S|〗^α 〗≥1. The least upper bound of all such α is defined as the distribution’s characteristic number. The useful extreme values of the shape factor for various distributions which are found numerically before, the Beta, Kumaraswamy, Weibull, and GB2 Distribution, are derived using asymptotic analysis. The match of the numerical and the analytical results can be considered prove of each other. The characteristic numbers of these distributions are also calculated. The study of the boundary value of the shape factor, or the shape factor asymptotic analysis, help reveal properties of the original shape factor, and reveal relationship between distributions, such as between the Kumaraswamy distribution and the Weibull distribution.