On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension
Authors:
T. Bodnar, S. Mazur, S. Muhinyuza and N. Parolya
Journal:
Theor. Probability and Math. Statist. 99 (2019), 39-52
MSC (2010):
Primary 60E05, 60E10, 60F05, 62H10, 62E20
DOI:
https://doi.org/10.1090/tpms/1078
Published electronically:
February 27, 2020
MathSciNet review:
3908654
Full-text PDF
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Additional Information
Abstract: In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation of this product is derived. Using this representation, characteristic function and asymptotic distribution of the product under the double asymptotic regime are established. We further document a good finite sample performance of the obtained high-dimensional asymptotic distribution via an extensive Monte Carlo study.
References
- Jose-M. Bernardo and Adrian F. M. Smith, Bayesian theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1994. MR 1274699
- Olha Bodnar, Sequential surveillance of the tangency portfolio weights, Int. J. Theor. Appl. Finance 12 (2009), no. 6, 797–810. MR 2590292, DOI https://doi.org/10.1142/S0219024909005464
- 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
- Taras Bodnar, Stepan Mazur, and Krzysztof Podgórski, Singular inverse Wishart distribution and its application to portfolio theory, J. Multivariate Anal. 143 (2016), 314–326. MR 3431434, DOI https://doi.org/10.1016/j.jmva.2015.09.021
- 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
- M. Britten-Jones, The sampling error in estimates of mean-variance efficient portfolio weights, J. Finance 54 (1999), 655–671.
- José A. Díaz-García, Ramón Gutierrez Jáimez, and Kanti V. Mardia, Wishart and pseudo-Wishart distributions and some applications to shape theory, J. Multivariate Anal. 63 (1997), no. 1, 73–87. MR 1491567, DOI https://doi.org/10.1006/jmva.1997.1689
- Geof H. Givens and Jennifer A. Hoeting, Computational statistics, 2nd ed., Wiley Series in Computational Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2013. MR 3236433
- 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
- J. D. Jobson and B. Korkie, Estimation for Markowitz efficient portfolios, J. Amer. Statist. Assoc. 75 (1980), no. 371, 544–554. MR 590686
- R. Kan and G. Zhou, Optimal portfolio choice with parameter uncertainty, J. Financ. Quant. Anal. 42 (2007), 621–656.
- Sujit Kumar Mitra, A density-free approach to the matrix variate beta distribution, Sankhyā Ser. A 32 (1970), 81–88. MR 293757
- I. Kotsiuba and S. Mazur, On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian vector, Teor. Ĭmovīr. Mat. Stat. 93 (2015), 95–104 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 93 (2016), 103–112. MR 3553443, DOI https://doi.org/10.1090/tpms/1004
- Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR 652932
- Dimitrios Pappas, Konstantinos Kiriakopoulos, and George Kaimakamis, Optimal portfolio selection with singular covariance matrix, Int. Math. Forum 5 (2010), no. 45-48, 2305–2318. MR 2727027
- Serge B. Provost and Edmund M. Rudiuk, The exact distribution of indefinite quadratic forms in noncentral normal vectors, Ann. Inst. Statist. Math. 48 (1996), no. 2, 381–394. MR 1405939, DOI https://doi.org/10.1007/BF00054797
- Alvin C. Rencher and William F. Christensen, Methods of multivariate analysis, 3rd ed., Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2012. MR 2962097
- Muni Shanker Srivastava and C. G. Khatri, An introduction to multivariate statistics, North-Holland, New York-Oxford, 1979. MR 544670
- 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
- Harald Uhlig, On singular Wishart and singular multivariate beta distributions, Ann. Statist. 22 (1994), no. 1, 395–405. MR 1272090, DOI https://doi.org/10.1214/aos/1176325375
References
- J. M. Bernardo and A. F. M. Smith, Bayesian theory, Wiley, Chichester, 1994. MR 1274699
- O. Bodnar, Sequential surveillance of the tangency portfolio weights, Internat. J. Theoret. Appl. Finance 12 (2009), 797–810. MR 2590292
- T. Bodnar, S. Mazur, and Y. Okhrin, On exact and approximate distributions of the product of the Wishart matrix and normal vector, J. Multivar. Anal. 122 (2013), 70–81. MR 3189308
- T. Bodnar, S. Mazur, and K. Podgórski, Singular inverse Wishart distribution and its application to portfolio theory, J. Multivar. Anal. 143 (2016), 314–326. MR 3431434
- T. Bodnar and Y. Okhrin, Properties of the singular, inverse and generalized inverse partitioned Wishart distributions, J. Multivar. Anal. 99 (2008), 2389–2405. MR 2463397
- T. Bodnar and Y. Okhrin, On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory, Scandinavian J. Statistics 38 (2011), 311–331. MR 2829602
- M. Britten-Jones, The sampling error in estimates of mean-variance efficient portfolio weights, J. Finance 54 (1999), 655–671.
- J. A. Díaz-García, R. G. Jáimez, and K. V. Mardia, Wishart and pseudo-Wishart distributions and some applications to shape theory, J. Multivar. Anal. 63 (1997), 73–87. MR 1491567
- G. H. Givens and J. A. Hoeting, Computational statistics, John Wiley & Sons, New York, 2012. MR 3236433
- A. Gupta and D. Nagar, Matrix Variate Distributions, Chapman and Hall/CRC, Boca Raton, 2000. MR 1738933
- A. Gupta, T. Varga, and T. Bodnar, Elliptically contoured models in statistics and portfolio theory, Springer, Berlin, 2013. MR 3112145
- J. D. Jobson and B. Korkie, Estimation for Markowitz efficient portfolios, J. Amer. Stat. Assoc. 75 (1980), 544–554. MR 590686
- R. Kan and G. Zhou, Optimal portfolio choice with parameter uncertainty, J. Financ. Quant. Anal. 42 (2007), 621–656.
- C. Khatri, A note on Mitra’s paper “A density free approach to the matrix variate beta distribution”, Sankhyā: Indian J. Statist., Ser. A 32 (1970), 311–318. MR 293757
- I. Kotsiuba and S. Mazur, On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a gaussian random vector, Theory of Probability and Mathematical Statistics 93 (2015), 95–104. MR 3553443
- R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982. MR 652932
- D. Pappas, K. Kiriakopoulos, and G. Kaimakamis, Optimal portfolio selection with singular covariance matrix, Internat. Math. Forum 5 (2010), 2305–2318. MR 2727027
- S. B. Provost and E. M. Rudiuk, The exact distribution of indefinite quadratic forms in noncentral normal vectors, Ann. Inst. Statist. Math. 48 (1996), 381–394. MR 1405939
- A. C. Rencher and W. F. Christensen, Methods of Multivariate Analysis, Third edition, Wiley Online Library, 2012. MR 2962097
- M. Srivastava and C. Khatri, An Introduction to Multivariate Statistics, North-Holland, New York, 1979. MR 544670
- M. S. Srivastava, Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003), 1537–1560. MR 2012825
- H. Uhlig, On singular Wishart and singular multivariate beta distributions, Ann. Statist. 22 (1994), 395–405. MR 1272090
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Additional Information
T. Bodnar
Affiliation:
Department of Mathematics, Stockholm University, Roslagsvägen 101, SE-10691 Stockholm, Sweden
Email:
taras.bodnar@math.su.se
S. Mazur
Affiliation:
Unit of Statistics, School of Business, Örebro University, SE-70182 Örebro, Sweden
Email:
stepan.mazur@oru.se
S. Muhinyuza
Affiliation:
Department of Mathematics, Stockholm University, Roslagsvägen 101, SE-10691 Stockholm, Sweden; and Department of Mathematics, College of Science and technology, University of Rwanda, P.O. Box 3900, Kigali-Rwanda
Email:
stanislas.muhinyuza@math.su.se
N. Parolya
Affiliation:
Institute of Statistics, Leibniz University of Hannover, D-30167 Hannover, Germany
Email:
parolya@statistik.uni-hannover.de
Keywords:
Singular Wishart distribution,
singular normal distribution,
stochastic representation,
high-dimensional asymptotics
Received by editor(s):
April 28, 2018
Published electronically:
February 27, 2020
Additional Notes:
The first and third authors appreciate the financial support of SIDA via the project 1683030302.
The second author acknowledges the financial support from the project “Models for macro and financial economics after the financial crisis” (Dnr: P18-0201) funded by the Jan Wallander and Tom Hedelius Foundation.
Article copyright:
© Copyright 2020
American Mathematical Society
