On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian vector

Kotsiuba, I.; Mazur, S.

doi:10.1090/tpms/1004

On the asymptotic and approximate distributions of the product of an inverse Wishart matrix and a Gaussian vector

Authors: I. Kotsiuba and S. Mazur
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 93 (2015).
Journal: Theor. Probability and Math. Statist. 93 (2016), 103-112
MSC (2010): Primary 62E17, 62E20
DOI: https://doi.org/10.1090/tpms/1004
Published electronically: February 7, 2017
MathSciNet review: 3553443
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the distribution of the product of an inverse Wishart random matrix and a Gaussian random vector. We derive its asymptotic distribution as well as a formula for its approximate density function which is based on the Gaussian integral and the third order Taylor expansion. Furthermore, we compare the asymptotic and approximate density functions with the exact density obtained by Bodnar and Okhrin (2011). The results obtained in the paper are confirmed by the numerical study.

1. Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun, Dover Publications, Inc., New York, 1966. MR 0208797
2. S. F. Arnold, Mathematical Statistics, Prentice-Hall, New Jersey, 1990.
3. J. Bai and S. Shi, Estimating high dimensional covariance matrices and its applications, Ann. Econom. Finance 12 (2011), 199-215.
4. Taras Bodnar and Arjun K. Gupta, Estimation of the precision matrix of a multivariate elliptically contoured stable distribution, Statistics 45 (2011), no. 2, 131–142. MR 2783401, https://doi.org/10.1080/02331880903541887
5. Taras Bodnar, Arjun K. Gupta, and Nestor Parolya, On the strong convergence of the optimal linear shrinkage estimator for large dimensional covariance matrix, J. Multivariate Anal. 132 (2014), 215–228. MR 3266272, https://doi.org/10.1016/j.jmva.2014.08.006
6. Taras Bodnar, Arjun K. Gupta, and Nestor Parolya, Direct shrinkage estimation of large dimensional precision matrix, J. Multivariate Anal. 146 (2016), 223–236. MR 3477661, https://doi.org/10.1016/j.jmva.2015.09.010
7. 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, https://doi.org/10.1016/j.jmva.2015.09.021
8. 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, https://doi.org/10.1016/j.jmva.2013.07.007
9. T. Bodnar, S. Mazur, and Y. Okhrin, Distribution of the product of singular Wishart matrix and normal vector, Teor. Ĭmovīr. Mat. Stat. 91 (2014), 1–14; English transl., Theory Probab. Math. Statist. 91 (2015), 1–15. MR 3364119, https://doi.org/10.1090/tpms/962
10. 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, https://doi.org/10.1016/j.jmva.2008.02.024
11. 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, https://doi.org/10.1111/j.1467-9469.2011.00729.x
12. Taras Bodnar and Wolfgang Schmid, A test for the weights of the global minimum variance portfolio in an elliptical model, Metrika 67 (2008), no. 2, 127–143. MR 2375302, https://doi.org/10.1007/s00184-007-0126-7
13. Tony Cai, Weidong Liu, and Xi Luo, A constrained ℓ₁ minimization approach to sparse precision matrix estimation, J. Amer. Statist. Assoc. 106 (2011), no. 494, 594–607. MR 2847973, https://doi.org/10.1198/jasa.2011.tm10155
14. T. Tony Cai and Ming Yuan, Adaptive covariance matrix estimation through block thresholding, Ann. Statist. 40 (2012), no. 4, 2014–2042. MR 3059075, https://doi.org/10.1214/12-AOS999
15. T. Tony Cai and Harrison H. Zhou, Minimax estimation of large covariance matrices under ℓ₁-norm, Statist. Sinica 22 (2012), no. 4, 1319–1349. MR 3027084
16. 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, https://doi.org/10.1006/jmva.1997.1689
17. Mathias Drton, Hélène Massam, and Ingram Olkin, Moments of minors of Wishart matrices, Ann. Statist. 36 (2008), no. 5, 2261–2283. MR 2458187, https://doi.org/10.1214/07-AOS522
18. David A. Harville, Matrix algebra from a statistician’s perspective, Springer-Verlag, New York, 1997. MR 1467237
19. Dietrich von Rosen, Moments for the inverted Wishart distribution, Scand. J. Statist. 15 (1988), no. 2, 97–109. MR 968156
20. P. Jorion, Bayes-Stein estimation for portfolio analysis, J. Financial Quant. Anal. 21 (1986), 279-292.
21. Olivier Ledoit and Michael Wolf, A well-conditioned estimator for large-dimensional covariance matrices, J. Multivariate Anal. 88 (2004), no. 2, 365–411. MR 2026339, https://doi.org/10.1016/S0047-259X(03)00096-4
22. Kantilal Varichand Mardia, John T. Kent, and John M. Bibby, Multivariate analysis, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York-Toronto, Ont., 1979. Probability and Mathematical Statistics: A Series of Monographs and Textbooks. MR 560319
23. Robb J. Muirhead, Aspects of multivariate statistical theory, John Wiley & Sons, Inc., New York, 1982. Wiley Series in Probability and Mathematical Statistics. MR 652932
24. Charles Stein, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. I, University of California Press, Berkeley and Los Angeles, 1956, pp. 197–206. MR 0084922
25. George P. H. Styan, Three useful expressions for expectations involving a Wishart matrix and its inverse, Statistical data analysis and inference (Neuchâtel, 1989) North-Holland, Amsterdam, 1989, pp. 283–296. MR 1089643, https://doi.org/10.1016/B978-0-444-88029-1.50032-0