The Valid Regions of Gram-Charlier Densities

40 Pages Posted: 24 Apr 2020

See all articles by Wei Lin

Wei Lin

Dept. Accountancy and Finance; College of mathematics and science

Jin E. Zhang

University of Otago, Otago Business School, Department of Accountancy and Finance

Date Written: March 31, 2020

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.

Keywords: Gram-Charlier Density; Edgeworth Series; Valid Regions; Higher Coefficients; Higher Cumulants.

JEL Classification: C40; C63, G13, G13

Suggested Citation

Lin, Wei and Zhang, Jin E., The Valid Regions of Gram-Charlier Densities (March 31, 2020). Available at SSRN: https://ssrn.com/abstract=3564891 or http://dx.doi.org/10.2139/ssrn.3564891

Wei Lin (Contact Author)

Dept. Accountancy and Finance ( email )

P.O. Box 56
Dunedin, Otago 9010
New Zealand
9054 (Fax)

College of mathematics and science ( email )

38 Zheda Road
Hangzhou, Zhejiang 310058
China

Jin E. Zhang

University of Otago, Otago Business School, Department of Accountancy and Finance ( email )

Dunedin, 9054
New Zealand
64 3 479 8575 (Phone)
64 3 479 8171 (Fax)

HOME PAGE: http://sites.google.com/site/jinzhanghomepage/home

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