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The Difference-of-Log-Normals Distribution: Properties, Estimation, and Growth

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  • Robert Parham

Abstract

This paper describes the Difference-of-Log-Normals (DLN) distribution. A companion paper makes the case that the DLN is a fundamental distribution in nature, and shows how a simple application of the CLT gives rise to the DLN in many disparate phenomena. Here, I characterize its PDF, CDF, moments, and parameter estimators; generalize it to N-dimensions using spherical distribution theory; describe methods to deal with its signature ``double-exponential'' nature; and use it to generalize growth measurement to possibly-negative variates distributing DLN. I also conduct Monte-Carlo experiments to establish some properties of the estimators and measures described.

Suggested Citation

  • Robert Parham, 2023. "The Difference-of-Log-Normals Distribution: Properties, Estimation, and Growth," Papers 2302.02486, arXiv.org.
    • Handle: RePEc:arx:papers:2302.02486
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    References listed on IDEAS

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    1. Archil Gulisashvili & Peter Tankov, 2013. "Tail behavior of sums and differences of log-normal random variables," Papers 1309.3057, arXiv.org, revised Jan 2016.
    2. Lorentz Jäntschi & Sorana D. Bolboacă, 2018. "Computation of Probability Associated with Anderson–Darling Statistic," Mathematics, MDPI, vol. 6(6), pages 1-17, May.
    3. C. F. Lo, 2012. "The Sum and Difference of Two Lognormal Random Variables," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, September.
    4. Marc F. Bellemare & Casey J. Wichman, 2020. "Elasticities and the Inverse Hyperbolic Sine Transformation," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 82(1), pages 50-61, February.
    5. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
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