TY - JOUR

T1 - Expectations of Ratios of Quadratic Forms in Normal Variables

T2 - Evaluating Some Top-Order Invariant Polynomials

AU - Smith, Murray D.

PY - 1993/9

Y1 - 1993/9

N2 - 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.

AB - 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.

KW - invariant polynomial

KW - Ratio of quadratic forms

KW - top‐order invariant polynomial

KW - top‐order zonal polynomial

KW - zonal polynomial

UR - http://www.scopus.com/inward/record.url?scp=84990519604&partnerID=8YFLogxK

U2 - 10.1111/j.1467-842X.1993.tb01335.x

DO - 10.1111/j.1467-842X.1993.tb01335.x

M3 - Article

AN - SCOPUS:84990519604

SN - 1026-597X

VL - 35

SP - 271

EP - 282

JO - Australian Journal of Statistics

JF - Australian Journal of Statistics

IS - 3

ER -