On Moments of Folded and Doubly Truncated Multivariate Extended Skew-Normal Distributions

This article develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle and the popular multivariate normal distribution. These recursions o...

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Bibliographic Details
Published in:Journal of computational and graphical statistics 2022-04, Vol.31 (2), p.455-465
Main Authors: Morales, Christian E. Galarza, Matos, Larissa A., Dey, Dipak K., Lachos, Victor H.
Format: Article
Language:English
Subjects:
Algorithms
Computation
Extended skew-normal distribution
Folded normal distribution
Integrals
Mathematical analysis
Multivariate analysis
Normal distribution
Product moments
Skewed distributions
Truncated distributions
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Summary:This article develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the truncated multivariate extended skew-normal distribution can be computed using a corresponding moment of a truncated multivariate normal distribution, pointing the way to a faster algorithm since a less number of integrals is required for its computation which results much simpler to evaluate. Since there are several methods available to calculate the first two moments of a multivariate truncated normal distribution, we propose an optimized method that offers a better performance in terms of time and accuracy, in addition to consider extreme cases in which other methods fail. Finally, we present an application in finance where multivariate tail conditional expectation (MTCE) for SN distributed data is calculated using analytical expressions involving normal left-truncated moments. The R MomTrunc package provides these new efficient methods for practitioners. Supplementary files for this article are available online.
ISSN:1061-8600
1537-2715
DOI:10.1080/10618600.2021.2000869