Some Theoretical and Computational Aspects of the Truncated Multivariate Skew-Normal/Independent Distributions

• Published in: Mathematics (Basel) Vol. 11; no. 16; p. 3579
• Main Authors: Morán-Vásquez, Raúl Alejandro, Zarrazola, Edwin, Nagar, Daya K.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2023
• Subjects: Affine transformations; Algorithms; Distribution (Probability theory); Food science; marginal distribution; Monte Carlo integration; Multivariate analysis; multivariate skew-normal/independent distributions; Normal distribution; Probability; Probability density functions; Probability theory; Random variables; random vector; Representations; Skewed distributions; Skewness; Software; Statistical analysis; truncated distribution
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11163579

In this article, we derive a closed-form expression for computing the probabilities of p-dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p-dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p-dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.