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

Authors:
Koshiro Yonenaga and Akio Suzukawa

Journal:
Theor. Probability and Math. Statist. **108** (2023), 209-224

MSC (2020):
Primary 62H10, 60E05

DOI:
https://doi.org/10.1090/tpms/1193

Published electronically:
May 2, 2023

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Abstract: We consider the distribution of the product of a Wishart matrix and a normal vector with uncommon covariance matrices. We derive the stochastic representation which reduces the computational burden for the generation of realizations of the product. Using this representation, the density function and higher order moments of the product are derived. In a numerical illustration, we investigate some properties of the distribution of the product. We further suggest the Edgeworth type expansions for the product, and we observe that the suggested approximations provide a good performance for moderately large degrees of freedom of a Wishart matrix.

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References
- D. Bauder, T. Bodnar, S. Mazur and Y. Okhrin,
*Bayesian inference for the tangent portfolio*, Int. J. Theor. Appl. Finance, **21** (2018), no. 8, 1850054, 27 pp. MR **3897158**
- T. Bodnar, S. Mazur, and Y. 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**
- T. Bodnar and Y. Okhrin,
*Properties of the singular, inverse and generalized inverse partitioned Wishart distributions*, J. Multivariate Anal. **99** (2008), no. 10, 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*, Scand. J. Stat. **38** (2011), no. 2, 311–331. MR **2829602**
- T. Bodnar, S. Mazur and Y. Okhrin,
*Distribution of the product of singular Wishart matrix and normal vector*, Theory Probab. Math. Statist. **91** (2015), 1–15. MR **3364119**
- T. Bodnar, S. Mazur and K. Podgórski,
*Singular inverse Wishart distribution and its application to portfolio theory*, J. Multivariate Anal. **143** (2016), 314–326. MR **3431434**
- T. Bodnar, S. Mazur, E. Ngailo and N. Parolya,
*Discriminant analysis in small and large dimensions*, Theory Probab. Math. Statist, **100** (2020), 21–41. MR **3992991**
- T. Bodnar, S. Mazur, S. Muhinzaya and N. Parolya,
*On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension*, Theory Probab. Math. Statist. **99** (2019), 39–52. MR **3908654**
- T. Bodnar, S. Mazur and N. Parolya,
*Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions*. Scand. J. Stat, **46** (2019), no. 2, 636–660. MR **3948571**
- J. A. Díaz-García, R. Gitiérrez-Jáimez and K. V. Mardia,
*Wishart and pseudo-Wishart distributions and some applications to shape theory*, J. Multivariate Anal. **63** (1997), no. 1, 73–87. MR **1491567**
- Y. Fujikoshi,
*Error bounds for asymptotic expansion of the distribution of the MLE in a GMANOVA model*, Ann. Inst. Statist. Math. **39** (1987), no. 1, 153–161. MR **886513**
- A. K. Gupta and D. K. Nagar,
*Matrix Variate Distributions*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 104, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR **1738933**
- A. K. Gupta, T. Varga, and T. Bodnar,
*Elliptically Contoured Models in Statistics and Portfolio Theory*, second edition, Springer, New York, 2013. MR **3112145**
- L. R. Haff,
*An identity for the Wishart distribution with applications*, J. Multivariate Anal. **9** (1979), no. 4, 531–544. MR **556910**
- P. D. Hoff,
*A first course in Bayesian statistical methods*, Springer Texts in Statistics, Springer, New York, 2009. MR **2648134**
- L. Isserlis,
*On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables*, Biometrika. **12** (1918), 134–139.
- F. Javed, S. Mazur and E. Ngailo,
*Higher order moments of the estimated tangency portfolio weights*, J. Appl. Stat. **48** (2021), no. 3, 517–535. MR **4205986**
- F. Javed, N. Loperdo and S. Mazur,
*Edgeworth expansions for multivariate random sums*. Econ. Stat. (2021), DOI 10.1016/j.ecosta.2021.04.005.
- R. Kan,
*From moments of sum to moments of product*, J. Multivariate Anal. **99** (2008), no. 3, 542–554. MR **2396978**
- 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 Probab. Math. Statist. **93** (2016), 103–112. MR **3553443**
- T. Kollo and D. von Rosen,
*Advanced Multivariate Statistics with Matrices*, Mathematics and Its Applications (New York), vol. 579, Springer, Dordrecht, 2005. MR **2162145**
- T. Kubokawa, M. Hyodo and M. S. Srivastava,
*Asymptotic expansion and estimation of EPMC for linear classification rules in high dimension*, J. Multivariate Anal. **115** (2013), 496–515. MR **3004572**
- G. Letac and H. Massam,
*All invariant moments of the Wishart distribution*, Scand. J. Statist. **31** (2004), no. 2, 295–318. MR **2066255**
- A. M. Mathai and S. B. Provost,
*Quadratic Forms in random variables*, Statistics: Textbooks and Monographs, vol. 126, Marcel Dekker, Inc., New York, 1992. MR **1192786**
- R. J. Muirhead,
*Aspects of multivariate statistical theory*, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1982. MR **652932**
- M. S. Srivastava,
*Singular Wishart and multivariate beta distributions*, Ann. Statist. **31** (2003), no. 5, 1537–1560. MR **2012825**
- D. von Rosen,
*Moments for the inverted Wishart distribution*, Scand. J. Stat. **15** (1988), no. 2, 97–109. MR **968156**
- D. von Rosen,
*Moments of maximum likelihood estimators in the growth curve model*, Statistics. **22** (1991), no. 1, 111–131. MR **1097365**
- D. von Rosen,
*Distribution and density approximations of a maximum likelihood estimator in the growth curve model*, Statistics. **29** (1997), no. 1, 37–59. MR **1438532**

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Additional Information

**Koshiro Yonenaga**

Affiliation:
Faculty of Education, Hokkaido University of Education, Sapporo, Japan

Email:
yonenaga.koshiro@s.hokkyodai.ac.jp

**Akio Suzukawa**

Affiliation:
Faculty of Economics and Business, Hokkaido University, Sapporo, Japan

Email:
suzukawa@econ.hokudai.ac.jp

Keywords:
Wishart distribution,
multivariate normal distribution,
moment

Received by editor(s):
June 25, 2021

Accepted for publication:
February 11, 2022

Published electronically:
May 2, 2023

Additional Notes:
Financial support from the Hokkaido University DX Doctoral Fellowship (JPMJSP2119) is gratefully acknowledged.

Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv