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

Bodnar, T.; Mazur, S.; Okhrin, Y.

doi:10.1090/tpms/962

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

Abstract | References | Similar Articles | 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

S. A. Andersson and G. G. Wojnar, Wishart distributions on homogeneous cones, J. Theoret. Probab. 17 (2004), no. 4, 781–818. MR 2105736, DOI https://doi.org/10.1007/s10959-004-0576-z
Taras Bodnar and Yarema Okhrin, Properties of the singular, inverse and generalized inverse partitioned Wishart distributions, J. Multivariate Anal. 99 (2008), no. 10, 2389–2405. MR 2463397, DOI https://doi.org/10.1016/j.jmva.2008.02.024
Taras Bodnar and Yarema Okhrin, On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory, Scand. J. Stat. 38 (2011), no. 2, 311–331. MR 2829602, DOI https://doi.org/10.1111/j.1467-9469.2011.00729.x
Taras Bodnar, Stepan Mazur, and Yarema Okhrin, On the exact and approximate distributions of the product of a Wishart matrix with a normal vector, J. Multivariate Anal. 122 (2013), 70–81. MR 3189308, DOI https://doi.org/10.1016/j.jmva.2013.07.007
Mathias Drton, Hélène Massam, and Ingram Olkin, Moments of minors of Wishart matrices, Ann. Statist. 36 (2008), no. 5, 2261–2283. MR 2458187, DOI https://doi.org/10.1214/07-AOS522
Dan Geiger and David Heckerman, A characterization of the bivariate Wishart distribution, Probab. Math. Statist. 18 (1998), no. 1, Acta Univ. Wratislav. No. 2045, 119–131. MR 1644057
Dan Geiger and David Heckerman, Parameter priors for directed acyclic graphical models and the characterization of several probability distributions, Ann. Statist. 30 (2002), no. 5, 1412–1440. MR 1936324, DOI https://doi.org/10.1214/aos/1035844981
A. K. Gupta and D. K. Nagar, Matrix variate distributions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1738933
Arjun K. Gupta, Tamas Varga, and Taras Bodnar, Elliptically contoured models in statistics and portfolio theory, 2nd ed., Springer, New York, 2013. MR 3112145
David A. Harville, Matrix algebra from a statistician’s perspective, Springer-Verlag, New York, 1997. MR 1467237
C. G. Khatri, On the mutual independence of certain statistics, Ann. Math. Statist. 30 (1959), 1258–1262. MR 110135, DOI https://doi.org/10.1214/aoms/1177706112
Hélène Massam and Jacek Wesołowski, The Matsumoto-Yor property and the structure of the Wishart distribution, J. Multivariate Anal. 97 (2006), no. 1, 103–123. MR 2208845, DOI https://doi.org/10.1016/j.jmva.2004.11.008
A. M. Mathai and Serge B. Provost, Quadratic forms in random variables, Statistics: Textbooks and Monographs, vol. 126, Marcel Dekker, Inc., New York, 1992. Theory and applications. MR 1192786
I. Olkin and S. N. Roy, On multivariate distribution theory, Ann. Math. Statistics 25 (1954), 329–339. MR 61795, DOI https://doi.org/10.1214/aoms/1177728789
Ingram Olkin and Herman Rubin, Multivariate beta distributions and independence properties of the Wishart distribution, Ann. Math. Statist. 35 (1964), 261–269. MR 160297, DOI https://doi.org/10.1214/aoms/1177703748
Alvin C. Rencher, Methods of multivariate analysis, 2nd ed., Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1885894
M. S. Srivastava, Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003), no. 5, 1537–1560. MR 2012825, DOI https://doi.org/10.1214/aos/1065705118

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 62H10, 60E05, 60E10

Retrieve articles in all journals with MSC (2010): 62H10, 60E05, 60E10

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