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

Hillier, G.H. (1999) The density of a quadratic form in a vector uniformly distributed on the n-sphere , Southampton, UK University of Southampton (Discussion Papers in Economics and Econometrics, (doi:10.1017/S026646660117101X) , 9902).

Record type: Monograph (Discussion Paper)


There are many instances in the statistical literature in which inference is based on a normalised quadratic form in a standard normal vector, normalised by the squared length of that vector. Examples include both test statistics (the Durbin-Watson statistic, and many other diagnostic test statistics for linear models), and estimators (serial correlation coefficients).

Although the properties of such a statistic have been much studied – particularly for the special case of serial correlation coefficients – its density function remains unknown. Two of the earliest contributors to this literature, von Neuman (1941) and Koopmans (1942), provided what are still today almost the entire extent of our knowledge of the density.

This paper gives formulae for the density function of such a statistic in each of the open intervals between the characteristic roots of the matrix involved. We do not assume that these roots are positive, but do assume that they are distinct. The case of non-distinct roots can be dealt with by methods similar to those used here. Starting from a representation of the density as a surface integral over an (n-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: 1999
Organisations: Economics


Local EPrints ID: 33137
PURE UUID: a711cf21-9070-4e98-9c03-ceecbc32a1d1

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Date deposited: 24 May 2007
Last modified: 17 Jul 2017 15:53

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