Scandinavian Working Papers in Business Administration

Working Papers,
Örebro University, School of Business

No 2026:7: On the Exact Distribution of the Product of an Inverse Wishart Matrix and a Normal Vector

Taras Bodnar (), Raymond Kan (), Stepan Mazur (), Jiening Pan () and Xiaolu Wang ()
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
Jiening Pan: Nankai University, China, Postal: School of Finance, Nankai University, Tianjin, 300350, China
Xiaolu Wang: Iowa State University, USA, Postal: Iowa State University, 2167 Union Drive, Ames, IA, 50011, USA

Abstract: We derive the exact density function of the product of an inverseWishart random matrix and an independent normal random vector. The density is expressed as a one-dimensional integral that can be evaluated quickly and accurately by standard quadrature methods. We further obtain integral representations for the density of linear combinations of the elements of this product; in particular, the joint density of any p linear combinations can be recovered from an integral of dimension at most min(p+1, m− p+1, 6), where m is the dimension of the random vector. All results are established in the general setting in which the scale matrix of the Wishart distribution and the covariance matrix of the normal vector are arbitrary positive definite matrices, and simplified formulas involving integrals of dimension at most three are provided for the important special case in which the two covariance matrices are proportional.

Keywords: Inverse Wishart distribution; multivariate normal distribution; exact distribution; integral representation; hypergeometric function; parabolic cylinder function

JEL-codes: C02; C10; C46; C63

Language: English

26 pages, August 26, 2026

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