TY - JOUR

T1 - Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum-Saunders distributions

AU - Caro-Lopera, Francisco J.

AU - Leiva, Víctor

AU - Balakrishnan, N.

N1 - Funding Information:
The authors wish to thank the Editor and two anonymous referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. The research work of F.J. Caro–Lopera was supported by the joint Grant 470 (SUMMA group) of the Universidad de Medellín, Colombia, and CIMAT , México. He also thanks Grant 105657 of CONACYT, México . The research work of V. Leiva was partially supported by Grant FONDECYT 1080326 from Chile. Finally, the research work of N. Balakrishnan was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada, and a grant (Project number KSU-VPP-107) from King Saud University (Riyadh, Saudi Arabia).

PY - 2012/2

Y1 - 2012/2

N2 - In this paper, we establish a connection between the Hadamard product and the usual matrix multiplication. In addition, we study some new properties of the Hadamard product and explore the inverse problem associated with the established connection, which facilitates diverse applications. Furthermore, we propose a matrix-variate generalized Birnbaum-Saunders (GBS) distribution. Three representations of the matrix-variate GBS density are provided, one of them by using the mentioned connection. The main motivation of this article is based on the fact that the representation of the matrix-variate GBS density based on element-by-element specification does not allow matrix transformations. Consequently, some statistical procedures based on this representation, such as multivariate data analysis and statistical shape theory, cannot be performed. For this reason, the primary goal of this work is to obtain a matrix representation of the matrix-variate GBS density that is useful for some statistical applications. When the GBS density is expressed by means of a matrix representation based on the Hadamard product, such a density is defined in terms of the original matrices, as is common for many matrix-variate distributions, allowing matrix transformations to be handled in a natural way and then suitable statistical procedures to be developed.

AB - In this paper, we establish a connection between the Hadamard product and the usual matrix multiplication. In addition, we study some new properties of the Hadamard product and explore the inverse problem associated with the established connection, which facilitates diverse applications. Furthermore, we propose a matrix-variate generalized Birnbaum-Saunders (GBS) distribution. Three representations of the matrix-variate GBS density are provided, one of them by using the mentioned connection. The main motivation of this article is based on the fact that the representation of the matrix-variate GBS density based on element-by-element specification does not allow matrix transformations. Consequently, some statistical procedures based on this representation, such as multivariate data analysis and statistical shape theory, cannot be performed. For this reason, the primary goal of this work is to obtain a matrix representation of the matrix-variate GBS density that is useful for some statistical applications. When the GBS density is expressed by means of a matrix representation based on the Hadamard product, such a density is defined in terms of the original matrices, as is common for many matrix-variate distributions, allowing matrix transformations to be handled in a natural way and then suitable statistical procedures to be developed.

KW - Generalized birnbaum-saunders distribution

KW - Kronecker product

KW - Multivariate analysis

KW - Schur or entry-wise product

KW - Shape theory

UR - http://www.scopus.com/inward/record.url?scp=80052267159&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2011.07.004

DO - 10.1016/j.jmva.2011.07.004

M3 - Article

AN - SCOPUS:80052267159

VL - 104

SP - 126

EP - 139

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -