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EXPECTATIONS OF RATIOS OF QUADRATIC FORMS IN NORMAL VARIABLES: EVALUATING SOME TOP‐ORDER INVARIANT POLYNOMIALS
Author(s) -
Smith Murray D.
Publication year - 1993
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1993.tb01335.x
Subject(s) - mathematics , polynomial matrix , invariant (physics) , matrix polynomial , quadratic equation , matrix (chemical analysis) , polynomial , order (exchange) , pure mathematics , mathematical analysis , materials science , geometry , finance , composite material , economics , mathematical physics
Summary Chikuse's (1987) algorithm constructs top‐order invariant polynomials with multiple matrix arguments. Underlying it is a set of simultaneous equations for which all integer solutions must be found. Each solution represents a component of the sum of terms which comprise the polynomial. The system of equations has a specialised structure which may be exploited to obtain a polynomial with r matrix arguments in terms of a polynomial with r ‐1 matrix arguments. This is demonstrated for two particular polynomials that have two matrix arguments. These results are applied to problems involving expectations of ratios of quadratic forme in normal variables; analytic as well as computable formulae are derived.