Shape Factor Asymptotic Analysis I
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.
 Hardy, G.H., Wright, E.M. 1979. An Introduction to the Theory of Numbers, 5th Edition Oxford, England: Clarendon Press, ISBN: 978-0-19-853171-5.
 Marichev, O., Trott, M. 2013. The Ultimate Univariate Probability Distribution Explorer. Available from: http://blog.wolfram.com/2013/02/01/the-ultimate-univariate-probability-distribution-explorer/
 McDonald, J.B. 1984. Some generalized functions for the size distribution of income. Econometrica, 52(3): 647-663.
 McDonald, J.B., Sorensen, J., Turley, P.A. 2011. Skewness and kurtosis properties of income distribution models. LIS Working Paper Series, No. 569. Review of Income and Wealth. http://dx.doi.org/10.1111/j.1475-4991.2011.00478.x
 McNeil, A.J., Frey, R., Embrechts, P. 2015. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, NJ, USA, ISBN: 978-0691166278.
 Wang, F.X. 2018. What determine EP curve shape? Available from: http://dx.doi.org/10.13140/RG.2.2. 30056.11523
 Wang, F.X. 2019. What determines EP curve shape? In: Bruno Carpentieri (Ed) Applied Mathematics. http://dx.doi.org/10.5772/intechopen.82832. Available from: https://cdn.intechopen.com/pdfs/64962.pdf
The Copyright Transfer Form to ASERS Publishing (The Publisher)
This form refers to the manuscript, which an author(s) was accepted for publication and was signed by all the authors.
The undersigned Author(s) of the above-mentioned Paper here transfer any and all copyright-rights in and to The Paper to The Publisher. The Author(s) warrants that The Paper is based on their original work and that the undersigned has the power and authority to make and execute this assignment. It is the author's responsibility to obtain written permission to quote material that has been previously published in any form. The Publisher recognizes the retained rights noted below and grants to the above authors and employers for whom the work performed royalty-free permission to reuse their materials below. Authors may reuse all or portions of the above Paper in other works, excepting the publication of the paper in the same form. Authors may reproduce or authorize others to reproduce the above Paper for the Author's personal use or for internal company use, provided that the source and The Publisher copyright notice are mentioned, that the copies are not used in any way that implies The Publisher endorsement of a product or service of an employer, and that the copies are not offered for sale as such. Authors are permitted to grant third party requests for reprinting, republishing or other types of reuse. The Authors may make limited distribution of all or portions of the above Paper prior to publication if they inform The Publisher of the nature and extent of such limited distribution prior there to. Authors retain all proprietary rights in any process, procedure, or article of manufacture described in The Paper. This agreement becomes null and void if and only if the above paper is not accepted and published by The Publisher, or is with drawn by the author(s) before acceptance by the Publisher.