Positive Definite and Unimodal Gauss-Hermite Expansion of ProbabilityDensity Functions

The approximation of Gauss'ian-like probability densiiy funetions (p.dJ.) byGauss-Hermite series of Gram.Charlier aod Edgeworth type, ,or by a Cornish-Fisher · expansion frequently violates two oonstrairits: the 'curves should be' positive definite, aod unimodaJ. A new stanI dardization allows the choice ofthe basic Oauss .. Hennite system such that these diffieulties are avoided. If the p.d~( is given, the expansion coemcients of the approximating · series may be computed by a relative least-squares error 'fit; U8ually, the two p.dJ. constraints are not viola ted. If the moments are known, the expansion coefficients are I' computed by a matrix inversion, where an iterative pro1''cedure adjusts the standardizing parameters according to the two p.dJ. constraints. The method is compared to the · approximation by a Pearson funelion.