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All Invariant Moments of the Wishart Distribution
Author(s) -
Letac Gérard,
Massam Hélène
Publication year - 2004
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2004.01-043.x
Subject(s) - wishart distribution , mathematics , invariant (physics) , inverse wishart distribution , eigenvalues and eigenvectors , inverse , pure mathematics , matrix (chemical analysis) , combinatorics , algebra over a field , statistics , multivariate statistics , mathematical physics , physics , geometry , materials science , composite material , quantum mechanics
. In this paper, we compute moments of a Wishart matrix variate U of the form ( Q ( U )) where Q ( u ) is a polynomial with respect to the entries of the symmetric matrix u , invariant in the sense that it depends only on the eigenvalues of the matrix u . This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U − 1 . For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U . This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.