Journal of Probability and Statistics
Volume 2012 (2012), Article ID 758975, 10 pages
Research Article

Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables

1Banco de Portugal, Avenida Almirante Reis 71, 1150-012 Lisboa, Portugal
2ISEG, Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal

Received 13 October 2011; Accepted 2 February 2012

Academic Editor: Chunsheng Ma

Copyright © 2012 Maximiano Pinheiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Marginal probability density and cumulative distribution functions are presented for multidimensional variables defined by nonsingular affine transformations of vectors of independent two-piece normal variables, the most important subclass of Ferreira and Steel's general multivariate skewed distributions. The marginal functions are obtained by first expressing the joint density as a mixture of Arellano-Valle and Azzalini's unified skew-normal densities and then using the property of closure under marginalization of the latter class.