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Computation of the exact density function of the product of a Wishart matrix and a normal vector

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Abstract

This paper derives the density function of the product of a Wishart matrix and a normal vector that are independently distributed. Unlike existing results, we allow the covariance matrix of the Wishart distribution and that of the normal vector to differ, and we do not assume them to be proportional. In this general setting, the density is shown to admit a two-dimensional integral representation. We further obtain the density of linear combinations of the product. For a single linear combination, it is again expressed as a two-dimensional integral whose integrand involves the modified Bessel function of the second kind, and the special case of identity covariance matrices is treated explicitly. All resulting expressions are straightforward to evaluate numerically, and the product defines a new family of matrix-variate mixtures of multivariate normal distributions.

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  • Bodnar, Taras & Kan, Raymond & Mazur, Stepan, 2026. "Computation of the exact density function of the product of a Wishart matrix and a normal vector," Working Papers 2026:4, Örebro University, School of Business.
    • Handle: RePEc:hhs:oruesi:2026_004
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    References listed on IDEAS

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    1. Kan, Raymond & Zhou, Guofu, 2007. "Optimal Portfolio Choice with Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 42(3), pages 621-656, September.
    2. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2013. "On the exact and approximate distributions of the product of a Wishart matrix with a normal vector," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 70-81.
    3. Taras Bodnar & Wolfgang Schmid, 2009. "Econometrical analysis of the sample efficient frontier," The European Journal of Finance, Taylor & Francis Journals, vol. 15(3), pages 317-335.
    4. Raymond Kan & Daniel R. Smith, 2008. "The Distribution of the Sample Minimum-Variance Frontier," Management Science, INFORMS, vol. 54(7), pages 1364-1380, July.
    5. Raymond Kan & Xiaolu Wang & Guofu Zhou, 2022. "Optimal Portfolio Choice with Estimation Risk: No Risk-Free Asset Case," Management Science, INFORMS, vol. 68(3), pages 2047-2068, March.
    6. Taras Bodnar & Yarema Okhrin, 2011. "On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(2), pages 311-331, June.
    7. Kozubowski, Tomasz J. & Mazur, Stepan & Podgórski, Krzysztof, 2025. "Matrix variate gamma distributions with unrestricted shape parameter," Journal of Multivariate Analysis, Elsevier, vol. 209(C).
    8. Schöne, Alexander & Schmid, Wolfgang, 2000. "On the Joint Distribution of a Quadratic and a Linear Form in Normal Variables," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 163-182, February.
    9. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
    10. Arjun K. Gupta & Tamas Varga & Taras Bodnar, 2013. "Elliptically Contoured Models in Statistics and Portfolio Theory," Springer Books, Springer, edition 2, number 978-1-4614-8154-6, January.
    11. Adelchi Azzalini, 2005. "The Skew‐normal Distribution and Related Multivariate Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 159-188, June.
    12. Bao, Yong & Kan, Raymond, 2013. "On the moments of ratios of quadratic forms in normal random variables," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 229-245.
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    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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