The generalized lognormal distribution and the Stieltjes moment problem

TL;DR

This paper investigates the moment problem for the generalized lognormal distribution, revealing how its moment determinacy depends on a shape parameter and providing classes of distributions when moments are indeterminate.

Contribution

It characterizes the moment (in)determinacy of the generalized lognormal distribution based on its shape parameter and introduces Stieltjes classes for indeterminate cases.

Findings

Moment determinacy depends on the shape parameter.

Some cases have moments of all orders but are indeterminate.

A bounded case is uniquely determined by its moments.

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 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.