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Distribution of the product of a singular Wishart matrix and a normal vector


Authors: T. Bodnar, S. Mazur and Y. Okhrin
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 91 (2014).
Journal: Theor. Probability and Math. Statist. 91 (2015), 1-15
MSC (2010): Primary 62H10, 60E05; Secondary 60E10
Published electronically: February 3, 2016
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Abstract: In this paper we derive a very useful formula for the stochastic representation of the product of a singular Wishart matrix with a normal vector. Using this result, the expressions of the density function as well as of the characteristic function are established. Moreover, the derived stochastic representation is used to generate random samples from the product which leads to a considerable improvement in the computation efficiency. Finally, we present several important properties of the singular Wishart distribution, like its characteristic function and distributional properties of the partitioned singular Wishart matrix.


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Additional Information

T. Bodnar
Affiliation: Department of Mathematics, Humboldt-University of Berlin, Unter den Linden 6, D-10099 Berlin, Germany
Email: bodnar@math.hu-berlin.de

S. Mazur
Affiliation: Department of Statistics, Lund University, P.O. Box 743, SE-22007 Lund, Sweden
Email: stepan.mazur@stat.lu.se

Y. Okhrin
Affiliation: Department of Statistics, University of Augsburg, Universitaetsstrasse 16, D-86159 Augsburg, Germany
Email: yarema.okhrin@wiwi.uni-augsburg.de

DOI: https://doi.org/10.1090/tpms/962
Keywords: Singular Wishart distribution, multivariate normal distribution, stochastic representation, characteristic function
Received by editor(s): August 9, 2013
Published electronically: February 3, 2016
Additional Notes: The authors appreciate the financial support of the German Science Foundation (DFG), projects BO3521/2-2 and OK103/1-2, “Wishart Processes in Statistics and Econometrics: Theory and Applications”. The first author was partly supported by the German Science Foundation (DFG) via the Research Unit 1735 “Structural Inference in Statistics: Adaptation and Efficiency”
Article copyright: © Copyright 2016 American Mathematical Society