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

Taras Bodnar (), Raymond Kan () and Stepan Mazur ()
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Taras Bodnar: Linköping University, Postal: Department of Management and Engineering, Linköping University, 58183 Linköping, Sweden, https://liu.se/en/employee/tarbo31
Raymond Kan: University of Toronto, Postal: Rotman School of Management, University of Toronto, M5S 3E6 Ontario, Canada, https://discover.research.utoronto.ca/16430-raymond-kan
Stepan Mazur: Örebro University School of Business, Postal: Örebro University, School of Business, SE - 701 82 ÖREBRO, Sweden, https://www.oru.se/english/employee/stepan_mazur

No 2026:4, Working Papers from Örebro University, School of Business

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.

Keywords: matrix mixture of distributions; stochastic representation; Wishart distribution; inverse Wishart distribution; multivariate normal distribution; modified Bessel function of the second kind (search for similar items in EconPapers)
JEL-codes: C02 C10 C46 C63 (search for similar items in EconPapers)
Pages: 19 pages
Date: 2026-06-25
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