Theory of Probability and Mathematical Statistics
On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension
T. Bodnar, S. Mazur, S. Muhinyuza, N. Parolya
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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 which its characteristic function and asymptotic distribution 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.
Keywords: Singular Wishart distribution, singular normal distribution, stochastic representation, high-dimensional asymptotics.
Bibliography: 1. J. M. Bernardo, A. F. M. Smith, Bayesian theory, Chichester: Wiley, 1994.
2. O. Bodnar, Sequential surveillance of the tangency portfolio weights, International Journal of Theoretical and Applied Finance, 12 (2009), 797–810.
3. T. Bodnar, S. Mazur, Y. Okhrin, On exact and approximate distributions of the product of the Wishart matrix and normal vector, Journal of Multivariate Analysis, 122 (2013), 70–81.
4. T. Bodnar, S. Mazur, K. Podgórski, Singular inverse Wishart distribution and its application to portfolio theory, Journal of Multivariate Analysis, 143 (2016), 314–326.
5. T. Bodnar, Y. Okhrin, Properties of the singular, inverse and generalized inverse partitioned Wishart distributions, Journal of Multivariate Analysis, 99 (2008), 2389– 2405.
6. T. Bodnar, 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.
7. M. Britten-Jones, The sampling error in estimates of mean-variance efficient portfolio weights, The Journal of Finance, 54 (1999), 655–671.
8. J. A. Dı́az-Garcı́a, R. G. Jáimez, K. V. Mardia, Wishart and pseudo-Wishart distributions and some applications to shape theory, Journal of Multivariate Analysis, 63 (1997), 73–87.
9. G. H. Givens, J. A. Hoeting, Computational statistics, John Wiley & Sons, 2012.
10. A. Gupta, D. Nagar, Matrix Variate Distributions, Chapman and Hall/CRC, Boca Raton, 2000.
11. A. Gupta, T. Varga, T. Bodnar, Elliptically contoured models in statistics and portfolio theory, Springer, 2013.
12. J. D. Jobson, B. Korkie, Estimation for Markowitz efficient portfolios, Journal of the American Statistical Association, 75 (1980), 544–554.
13. R. Kan, G. Zhou, Optimal portfolio choice with parameter uncertainty, Journal of Financial and Quantitative Analysis, 42 (2007), 621–656.
14. C. Khatri, A note on Mitra’s paper ”A density free approach to the matrix variate beta distribution”, Sankhyā: The Indian Journal of Statistics, Series A, 32 (1970), 311–318.
15. I. Kotsiuba, 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.
16. R. J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982.
17. D. Pappas, K. Kiriakopoulos, G. Kaimakamis, Optimal portfolio selection with singular covariance matrix, International Mathematical Forum, 5 (2010), 2305–2318.
18. S. B. Provost, E. M. Rudiuk, The exact distribution of indefinite quadratic forms in noncentral normal vectors, Annals of the Institute of Statistical Mathematics, 48 (1996), 381–394.
19. A. C. Rencher, W. F. Christensen, Methods of multivariate analysis, third edition, Wiley Online Library, 2012.
20. M. Srivastava, C. Khatri, An introduction to multivariate statistics, North-Holland, New York, 1979.
21. M. S. Srivastava, Singular Wishart and multivariate beta distributions, The Annals of Statistics, 31 (2003), 1537–1560.
22. H. Uhlig, On singular Wishart and singular multivariate beta distributions, The Annals of Statistics, 22 (1994), 395–405.
