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Shape Factor Asymptotic Analysis I

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  • Wang, Frank Xuyan

Abstract

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.

Suggested Citation

  • Wang, Frank Xuyan, 2019. "Shape Factor Asymptotic Analysis I," MPRA Paper 93357, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:93357
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    File URL: https://mpra.ub.uni-muenchen.de/93357/1/MPRA_paper_93357.pdf
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    References listed on IDEAS

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    1. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    2. James B. Mcdonald & Jeff Sorensen & Patrick A. Turley, 2013. "Skewness And Kurtosis Properties Of Income Distribution Models," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 59(2), pages 360-374, June.
    3. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
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    More about this item

    Keywords

    Shape Factor; Skewness; Kurtosis; Asymptotic Expansion; Beta Distribution; Kumaraswamy Distribution; Weibull Distribution; GB2 Distribution; Computer Algebra System; Numerical Optimization; Characteristic Number.;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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