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The Generalized Lognormal Distribution and the Stieltjes Moment Problem

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  • Christian Kleiber

    (Universität Basel)

Abstract

This paper studies a Stieltjes-type moment problem defined by the generalized lognormal distribution, a heavy-tailed distribution with applications in economics, finance, and related fields. It arises as the distribution of the exponential of a random variable following a generalized error distribution, and hence figures prominently in the exponential general autoregressive conditional heteroskedastic (EGARCH) model of asset price volatility. Compared to the classical lognormal distribution it has an additional shape parameter. It emerges that moment (in)determinacy depends on the value of this parameter: for some values, the distribution does not have finite moments of all orders, hence the moment problem is not of interest in these cases. For other values, the distribution has moments of all orders, yet it is moment-indeterminate. Finally, a limiting case is supported on a bounded interval, and hence determined by its moments. For those generalized lognormal distributions that are moment-indeterminate, Stieltjes classes of moment-equivalent distributions are presented.

Suggested Citation

  • Christian Kleiber, 2014. "The Generalized Lognormal Distribution and the Stieltjes Moment Problem," Journal of Theoretical Probability, Springer, vol. 27(4), pages 1167-1177, December.
    • Handle: RePEc:spr:jotpro:v:27:y:2014:i:4:d:10.1007_s10959-013-0477-0
    DOI: 10.1007/s10959-013-0477-0
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    References listed on IDEAS

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    1. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-370, March.
    2. Samuel Kotz & Christian Kleiber, 2002. "A characterization of income distributions in terms of generalized Gini coefficients," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 789-794.
    3. Alfarano, Simone & Milaković, Mishael & Irle, Albrecht & Kauschke, Jonas, 2012. "A statistical equilibrium model of competitive firms," Journal of Economic Dynamics and Control, Elsevier, vol. 36(1), pages 136-149.
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    Cited by:

    1. Simon P Anderson & André de Palma, 2020. "Decoupling the CES Distribution Circle with Quality and Beyond: Equilibrium Distributions and the CES-Logit Nexus," The Economic Journal, Royal Economic Society, vol. 130(628), pages 911-936.
    2. Gwo Dong Lin, 2017. "Recent developments on the moment problem," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-17, December.
    3. Rafael Sala Mayato & Patrick Loughlin & Leon Cohen, 2022. "Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1537-1555, September.

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    More about this item

    Keywords

    Generalized error distribution; Generalized lognormal distribution; Lognormal distribution; Moment problem; Size distribution; Stieltjes class; Volatility model;
    All these keywords.

    JEL classification:

    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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