Asymptotic theory for statistics based on cumulant vectors with applications

• Published in: Scandinavian journal of statistics Vol. 48; no. 2; pp. 708 - 728
• Main Authors: Rao Jammalamadaka, Sreenivasa, Taufer, Emanuele, Terdik, György H.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.06.2021
• Subjects: Asymptotic properties; Commutators; elliptically symmetric distributions; Gram–Charlier expansion; Independent component analysis; Kurtosis; Mathematical analysis; Multivariate analysis; multivariate cumulants; multivariate skewness and kurtosis; Skewed distributions; Skewness
• ISSN: 0303-6898; 1467-9469
• DOI: 10.1111/sjos.12521

For any given multivariate distribution, explicit formulae for the asymptotic covariances of cumulant vectors of the third and the fourth order are provided here. General expressions for cumulants of elliptically symmetric multivariate distributions are also provided. Utilizing these formulae one can extend several results currently available in the literature, as well as obtain practically useful expressions in terms of population cumulants, and computational formulae in terms of commutator matrices. Results are provided for both symmetric and asymmetric distributions, when the required moments exist. New measures of skewness and kurtosis based on distinct elements are discussed, and other applications to independent component analysis and testing are considered.