The Linearization of the Product of Two Zonal Polynomials

Kushner, H. B.

doi:10.1137/0519049

Cited by 12 publications

(6 citation statements)

References 18 publications

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“…Another statement and short proof may be found in Macdonald's (2013) notes. 9 Lemma 6 enables a direct evaluation of the c κ,α , but we also require the following result on the so-called linearization of a product of zonal polynomials (Constantine, 1966;Kushner, 1988;Macdonald, 1995). This says that, for α j, λ l, and certain coefficients g δ α,λ , we have…”

Section: Direct Approach Noncentral Casementioning

confidence: 99%

“…Finally, we need to compute the linearization coefficients g δ α,λ as defined in (52). These coefficients have been extensively studied (see, e.g., Kushner, 1988), and there are combinatorial formulas available for the linearization coefficients for products of the more general class of Jack polynomials (Stanley, 1989;Macdonald, 1995), of which the zonal polynomials are a special case. However, when D −1 j , D −1 l , and D j+l are available, we can simply use Theorem 3.1 of Richards (1982) to compute g δ α,λ with α j, λ l, and δ j + l as…”

Section: Transition Matrices and Computation Issuesmentioning

confidence: 99%

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Properties of the Inverse of a Noncentral Wishart Matrix

Hillier

Kan

2021

Econom. Theory

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The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.

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Section: Direct Approach Noncentral Casementioning

confidence: 99%

Section: Transition Matrices and Computation Issuesmentioning

confidence: 99%

Properties of the Inverse of a Noncentral Wishart Matrix

Hillier

Kan

2021

Econom. Theory

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“…However, it requires a considerable amount of calculation time to expand the zonal polynomials to large degrees. The following lemma is a formula for the special case of g δ κ,τ given by Kushner (1988).…”

Section: Preliminarymentioning

confidence: 99%

Numerical computation for the exact distribution of Roy's largest root statistic under linear alternative

Shimizu¹,

Hashiguchi²

2022

Preprint

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This paper discusses the computation of exact powers for Roy's test in multivariate analysis of variance (MANOVA). We derive an exact expression for the largest eigenvalue of a singular noncentral Beta matrix in terms of the product of zonal polynomials. The numerical computation for that distribution is conducted by an algorithm that expands the product of zonal polynomials as a linear combination of zonal polynomials. Furthermore, we provide an exact distribution of the largest eigenvalue in a form that is convenient for numerical calculations under the linear alternative.

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“…The coefficients g;o up to order k = 7 have been tabulated by Khatri and Pillai (1968). Kushner (1988) provides a general formula for g" (,~)o' LEMMA 3.1. Let U = Diag (ul, u2,..., ur) and g(U) be a symmetric function of ul~ u2~..., Ur,then (3.11) ...…”

Section: X-' [(K + T) -(Kj -T)] (38)mentioning

confidence: 99%