The density of a quadratic form in a vector uniformly distributed on the n-sphere
The density of a quadratic form in a vector uniformly distributed on the n-sphere
There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n [minus sign] 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.
1-28
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
2001
Hillier, Grant
3423bd61-c35f-497e-87a3-6a5fca73a2a1
Hillier, Grant
(2001)
The density of a quadratic form in a vector uniformly distributed on the n-sphere.
Econometric Theory, 17 (1), .
(doi:10.1017/S026646660117101X).
Abstract
There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n [minus sign] 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.
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Published date: 2001
Organisations:
Economics
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Local EPrints ID: 33410
URI: http://eprints.soton.ac.uk/id/eprint/33410
PURE UUID: 5c3105b2-2c9c-4468-ac34-012ee4a680fa
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Date deposited: 16 May 2006
Last modified: 16 Mar 2024 02:42
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