Analytic expressions for the positive definite and unimodal regions of Gram-Charlier series

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.