The density of a quadratic form in a vector uniformly distributed on the n-sphere

Hillier, Grant (2001) The density of a quadratic form in a vector uniformly distributed on the n-sphere. Econometric Theory, 17, (1), 1-28. (doi:10.1017/S026646660117101X).


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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.

Item Type: Article
Digital Object Identifier (DOI): doi:10.1017/S026646660117101X
ISSNs: 0266-4666 (print)
Related URLs:
Subjects: H Social Sciences > HA Statistics
Divisions : University Structure - Pre August 2011 > School of Social Sciences > Economics
ePrint ID: 33410
Accepted Date and Publication Date:
Date Deposited: 16 May 2006
Last Modified: 31 Mar 2016 11:59

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