Scandinavian Working Papers in Business Administration

Working Papers,
Örebro University, School of Business

No 2026:4: Computation of the exact density function of the product of a Wishart matrix and a normal vector

Taras Bodnar (), Raymond Kan () and Stepan Mazur ()
Additional contact information
Taras Bodnar: Linköping University, Postal: Department of Management and Engineering, Linköping University, 58183 Linköping, Sweden
Raymond Kan: University of Toronto, Postal: Rotman School of Management, University of Toronto, M5S 3E6 Ontario, Canada
Stepan Mazur: Örebro University School of Business, Postal: Örebro University, School of Business, SE - 701 82 ÖREBRO, Sweden

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

JEL-codes: C02; C10; C46; C63

Language: English

19 pages, June 25, 2026

Full text files 

wp-4-2026.pdf PDF-file Full text

Download statistics

Questions (including download problems) about the papers in this series should be directed to ()
Report other problems with accessing this service to Sune Karlsson ().

RePEc:hhs:oruesi:2026_004This page generated on 2026-07-09 04:38:46.