The Generalized Lognormal Distribution and the Stieltjes Moment Problem
The Generalized Lognormal Distribution and the Stieltjes Moment Problem
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广义对数正态分布和 Stieltjes 矩问题
DOI：
10.1007/s10959-013-0477-0
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发表时间：
2013
影响因子：
0.8
通讯作者：
Christian Kleiber
中科院分区：
文献类型：
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作者：
Christian Kleiber
本文研究了广义对数正态分布定义的Stieltjes型矩问题，广义对数正态分布是一种重尾分布，在经济、金融及相关领域有广泛的应用。它是作为服从广义误差分布的随机变量的指数分布出现的，因此在资产价格波动的指数广义自回归条件异方差(EGARCH)模型中占有重要地位。与经典的对数正态分布相比，它有一个额外的形状参数。结果表明，矩的确定性依赖于该参数的值：对于某些值，分布不具有所有阶数的有限阶矩，因此在这些情况下矩问题并不重要。对于其他值，该分布具有所有阶矩，但它是不确定的。最后，极限情形被支持在有界区间上，因此由它的矩决定。对于矩不确定的广义对数正态分布，给出了矩等价分布的Stieltjes类。
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.