The Valid Regions of Gram-Charlier Densities

Abstract

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by the truncated Gram-Charlier series where skewness and kurtosis directly appear as parameters. However, the existing literature is restricted to truncating the series expansion until the fourth moment because it becomes difficult to keep its non-negativity. This paper shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a moment of arbitrary even order represents a valid probability density. Furthermore, the impact of higher moments on the valid regions has been shown.