Shape Factor Asymptotic Analysis I

Abstract

We proposed using shape factor to distinguish probability distributions, and using relative minimum or maximum values of shape factor to locate distribution parameter allowable ranges for distribution fitting in our previous study. In this paper, the shape factor asymptotic analysis is employed to study such conditional minimum or maximum, to cross validate results found from numerical study and empirical formula we obtained and published earlier. The shape factor defined as kurtosis divided by skewness squared is characterized as the unique maximum choice of among all factors that is greater than or equal to 1 for all probability distributions. For all distributions from a specific distribution family, there may exists such that. The least upper bound of all such is defined as the distribution family’s characteristic number. The useful extreme values of the shape factor for various distributions that are found numerically before, the Beta, Kumaraswamy, Weibull, and GB2 distributions are derived using asymptotic analysis. The match of the numerical and the analytical results may arguably be considered proof of each other. The characteristic numbers of these distributions are also calculated. The study of the extreme 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 that between the Kumaraswamy distribution and the Weibull distribution.

Shape Factor Asymptotic Analysis II

“…We can similarly summarize our finding in Wang (2020) about Kumaraswamy distribution and Weibull distribution by saying that they are the GB1 distribution at…”

“…The trick in Equation ( 12) of Wang (2020) can be used to find whether an asymptotic limit of power function exist, and if exist, what the power order is. We used the following formula in Wang (2020) section 6.3 for GB2, and it will be used for GB1 as well:…”

“…This is the continuation of the study in Wang (2020) for shape factor, which is defined in Wang (2019a) with a plan of addressing probability distribution fitting validation and explanation problem by examining shape factor global and conditional minimum or maximum formulas or curves, to other distribution families. The problems include, to name a few, why it is not able to fit a data distribution by some distribution families, and why different distribution families can fit a data distribution almost as good as each other while their PDF or even distribution domains are very different.…”

“…The same symbolic and graphical techniques, as used in Wang (2020), will be applied to GB1 in this paper, THT distribution (defined in Wang, 2019b), GH distribution and other families later, for getting the boundary forms of the shape factor. The global minimum value of the shape factor will serve as the limit or bound of the conditional minimum value of the shape factor for some given parameters while keep other parameters unchanged.…”

“…We will refer freely on contents of Wang (2019a and2020) rather than repeat it here, to avoid self-copying. Distributions definitions and parameter assumptions will refer to Marichev and Trott (2013).…”