Moments and Identities Involving Inverted Wishart Distribution RAJA MOHAMMAD LATIF Department of Mathematics and Natural Sciences Prince Mohammad Bin Fahd University P.O. Box 1664 Al Khobar KINGDOM OF SAUDI ARABIA rlatif@pmu.edu.sa; rajamlatif@gmail.com & dr.rajalatif@yahoo.com https://www.pmu.edu.sa/profiles/rlatif/home ANWAR H. JOARDER Faculty of Science and Engineering Northern University of Business and Technology Khulna BANGLADESH ajstat@gmail.com Abstract: - Moments of multivariate Wishart Distribution are known up to fourth order. But in many contexts, moments of some functions of Wishart distribution and Inverted Wishart Distribution have been found useful in risk theoretic estimation of covariance matrix and its characteristics. In this paper we review moments of some important functions of Wishart and Inverted Distributions. Key-Words: - Wishert distribution, Inverted Wishert distribution, Covariance matrix, Wishert matrix, Jacobian, Moments Received: February 13, 2020. Revised: April 20, 2020. Accepted: April 29, 2020. Published: April 30, 2020. 1 Introduction The inverted Wishart distribution has got various applications in statistics. For instance, the distribution can be used as a natural conjugate prior when dealing with Bayesian estimation of covariance matrix under sampling from multivariate normal distribution (Anderson, 2003, Section 7.7). Moments of the inverted Wishart distribution have also been utilized in discriminant analysis (Das Gupta, 1968; Siskind, 1972 and Haff, 1982) while obtaining moments of the maximum likelihood estimators in the growth curve model (von Rosen, 1988 and Von Rosen, 1997). Useful results for the inverted Wishart distribution can also be found in Press (1982). Kaufman (1967) derived the moments by factorization theorem. Das Gupta (1968) utilized some invariance arguments. Both of them based their calculations on the inverse moments of chi- square variable. In a series of papers, Haff (1977, 1979, 1980, 1981, 1982) presented moment identities which are useful for deriving moments of inverted Wishart distribution. The identities were established by applying Stokes’ theorem. Independently of Haff, von Rosen (1985) derived moments of inverted Wishart distribution with the help of matrix calculus. These moments have been found useful in risk theoretic estimation of covariance matrix and its characteristics. See for example Joarder (1997) and Joarder (1998). Let 1 2 , , ( ) N X X X N p  be a p -dimensional independent normal random vector with mean vector X so that the sums of squares and cross product matrix is given by 1 ( )( ) N j j j X X X X A       . Fisher (1915) derived the distribution of A for 2 p  in order to study the distribution of correlation coefficient from a normal sample. Wishart (1928) obtained the distribution for arbitrary p as the joint distribution of sample variances and covariances from multivariate normal population. Because of its important role in multivariate statistical analysis, various authors have derived it from different perspectives. See the references in Gupta and Nagar (2000, 87-88). The WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.14 Raja Mohammad Latif, Anwar H. Joarder E-ISSN: 2224-2880 139 Volume 19, 2020