Distribution of the product of a singular Wishart matrix and a normal vector
Authors:
T. Bodnar, S. Mazur and Y. Okhrin
Journal:
Theor. Probability and Math. Statist. 91 (2015), 1-15
MSC (2010):
Primary 62H10, 60E05; Secondary 60E10
DOI:
https://doi.org/10.1090/tpms/962
Published electronically:
February 3, 2016
MathSciNet review:
3364119
Full-text PDF Free Access
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Additional Information
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.
References
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References
- S. A. Andersson and G. G. Wojnar, Wishart distribution on homogeneous cones, Journal of Theoretical Probability 17 (2004), 781–818. MR 2105736 (2005k:62026)
- T. Bodnar and Y. Okhrin, Properties of the partitioned singular, inverse and generalized Wishart distributions, Journal of Multivariate Analysis 99 (2008), 2389–2405. MR 2463397 (2010b:62191)
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- T. Bodnar, S. Mazur, and Y. Okhrin, On the exact and approximate distributions of the product of a Wishart matrix with a normal vector, Journal of Multivariate Analysis 122 (2013), 70–81. MR 3189308
- M. Drton, H. Massam, and I. Olkin, Moments of minors of Wishart matrices, Annals of Statistics 36 (2008), 2261–2283. MR 2458187 (2010a:60035)
- D. Geiger and D. Heckerman, A characterization of the bivariate Wishart distribution, Probability and Mathemathical Statistics 18 (1998), 119–131. MR 1644057 (2000a:62122)
- D. Geiger and D. Heckerman, Parameter priors for directed acyclic graphical models and the characterization of several probability distributions, Annals of Statistics 30 (2002), 1412–1440. MR 1936324 (2003h:62019)
- A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman and Hall/CRC, Boca Raton, 2000. MR 1738933 (2001d:62055)
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- D. A. Harville, Matrix Algebra from Statistician’s Perspective, Springer, New York, 1997. MR 1467237 (98k:15001)
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- M. S. Srivastava, Singular Wishart and multivariate beta distributions, Annals of Statistics 31 (2003), 1537–1560. MR 2012825 (2004g:62106)
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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
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