Abstract

It often arises in practice that, although the first few moments of a distribution are known, the density of the distribution cannot be determined in closed form. In such cases, Gram-Charlier and Edgeworth series are commonly used to analytically approximate the unknown density in terms of the known moments. Although convenient, these series contain polynomial factors, and can hence lead to density approximations taking negative values or becoming multimodal in general. Consequently it is of interest to determine the set of moments for which the corresponding density approximations are positive definite and unimodal. In contrast to the existing literature that determines the boundaries of such sets numerically, explicit analytic expressions for the two boundaries are given in this paper for the Gram-Charlier series. Moreover, a method for projecting a given set of moments onto the boundaries of the two regions in order to minimizes the Kolmogorov-Smirnov statistic of corresponding density approximations is also provided.

摘要

在实践中经常出现这样的情况，虽然分布的前几个矩是已知的，但分布的密度不能以封闭形式确定。在这种情况下，Gram-Charlier 和 Edgeworth 级数通常用于根据已知矩分析近似未知密度。虽然方便，但这些级数包含多项式因子，因此可能导致密度近似取负值或一般成为多峰。因此，确定相应的密度近似为正定且单峰的矩集是有意义的。与现有文献以数值方式确定此类集合的边界相比，本文针对 Gram-Charlier 系列给出了两个边界的明确解析表达式。而且，