Shape factor asymptotic analysis II

Probability distributions with identical shape factor asymptotic limit formulas are defined as asymptotic equivalent distributions. The GB1, GB2, and Generalized Gamma distributions are examples of asymptotic equivalent distributions, which have similar fitting capabilities to data distribution with comparable parameters values. These example families are also asymptotic equivalent to Kumaraswamy, Weibull, Beta, ExpGamma, Normal, and LogNormal distributions at various parameters boundaries. The asymptotic analysis that motivated the asymptotic equivalent distributions definition is further generalized to contour analysis, with contours not necessarily parallel to the axis. Detailed contour analysis is conducted for GB1 and GB2 distributions for various contours of interest. Methods combing induction and symbolic deduction are crafted to resolve the dilemma over conflicting symbolic asymptotic limit results. From contour analysis build on graphical and analytical reasoning, we find that the upper bound of the GB2 distribution family, having the maximum shape factor for given skewness, is the Double Pareto distribution.