A revisit of the Gram-Charlier and Edgeworth series expansions

Abstract

In this paper we make several observations on the Gram- Charlier and Edgeworth series, which are methods for modeling and approximating probability density functions.We present a simplified derivation which highlights both the similarity and the differences of the series expansions, that are often obscured by alternative derivations. We also introduce a reformulation of the Edgeworth series in terms of the complete exponential Bell polynomials, which make both series easy to implement and evaluate. The result is a significantly more accessible methodology, in the sense that it is easier to understand and to implement. Finally, we also make a remark on the Gram-Charlier series with a gamma kernel, providing a novel and simple expression for its coefficients.