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

DOI:
https://doi.org/10.1090/tpms/962

Published electronically:
February 3, 2016

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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.

**1.**S. A. Andersson and G. G. Wojnar,*Wishart distribution on homogeneous cones*, Journal of Theoretical Probability**17**(2004), 781-818. MR**2105736 (2005k:62026)****2.**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)****3.**T. Bodnar and Y. Okhrin,*On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory*, Scandinavian Journal of Statistics**38**(2011), 311-331. MR**2829602 (2012e:62172)****4.**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****5.**M. Drton, H. Massam, and I. Olkin,*Moments of minors of Wishart matrices*, Annals of Statistics**36**(2008), 2261-2283. MR**2458187 (2010a:60035)****6.**D. Geiger and D. Heckerman,*A characterization of the bivariate Wishart distribution*, Probability and Mathemathical Statistics**18**(1998), 119-131. MR**1644057 (2000a:62122)****7.**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)****8.**A. K. Gupta and D. K. Nagar,*Matrix Variate Distributions*, Chapman and Hall/CRC, Boca Raton, 2000. MR**1738933 (2001d:62055)****9.**A. K. Gupta, T. Varga, and T. Bodnar,*Elliptically Contoured Models in Statistics and Portfolio Theory*, Springer, New York, 2013. MR**3112145****10.**D. A. Harville,*Matrix Algebra from Statistician's Perspective*, Springer, New York, 1997. MR**1467237 (98k:15001)****11.**C. G. Khatri,*On the mutual independence of certain statistics*, Annals of Mathematical Statistics**30**(1959), 1258-1262. MR**0110135 (22:1017)****12.**H. Massam and J. Weseĺowski,*The Matsumoto-Yor property and the structure of the Wishart distribution*, Journal of Multivariate Analysis**97**(2006), 103-123. MR**2208845 (2007f:62040)****13.**A. M. Mathai and S. B. Provost,*Quadratic Forms in Random Variables*, Marcel Dekker, Inc, New York, 1992. MR**1192786 (94g:62110)****14.**I. Olkin and S. N. Roy,*On multivariate distribution theory*, Annals of Mathematical Statistics**25**(1954), 329-339. MR**0061795 (15:885h)****15.**I. Olkin and H. Rubin,*Multivariate beta distribution and independency properties of Wishart distribution*, Annals of Mathematical Statistics**35**(1964), 261-269. MR**0160297 (28:3511)****16.**A. C. Rencher,*Methods of Multivariate Analysis*, A Wiley-Interscience publication, 2002. MR**1885894 (2003a:62004)****17.**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

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