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.
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Notes
It should be noted that these works employ different parameterizations of the distribution. Also, Nelson [17] obtains expectations of somewhat more general objects. Setting \(\gamma = 0,\,p=0\), and \(\theta =1\) in his Theorem A1.2 yields the required moments. The resulting expressions can be shown to coincide with those presented by Brunazzo and Pollastri [4].
This question was raised by an anonymous reviewer.
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Acknowledgments
I am grateful to Thomas Zehrt for helpful discussions and to an anonymous reviewer for a careful reading of an earlier draft.
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Kleiber, C. The Generalized Lognormal Distribution and the Stieltjes Moment Problem. J Theor Probab 27, 1167–1177 (2014). https://doi.org/10.1007/s10959-013-0477-0
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DOI: https://doi.org/10.1007/s10959-013-0477-0
Keywords
- Generalized error distribution
- Generalized lognormal distribution
- Lognormal distribution
- Moment problem
- Size distribution
- Stieltjes class
- Volatility model