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THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE

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  • Hillier, Grant
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    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 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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    Bibliographic Info

    Article provided by Cambridge University Press in its journal Econometric Theory.

    Volume (Year): 17 (2001)
    Issue (Month): 01 (February)
    Pages: 1-28

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    Handle: RePEc:cup:etheor:v:17:y:2001:i:01:p:1-28_17

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    Cited by:
    1. Peter M Robinson & Francesca Rossi, 2013. "Improved Tests for Spatial Correlation," STICERD - Econometrics Paper Series /2013/565, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    2. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Computationally efficient recursions for top-order invariant polynomials with applications," CeMMAP working papers CWP07/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Giovanni Forchini, . "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
    4. Zeng-Hua Lu & Maxwell King, 2002. "Improving The Numerical Technique For Computing The Accumulated Distribution Of A Quadratic Form In Normal Variables," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 149-165.
    5. Grant Hillier & Federico Martellosio, 2004. "Spatial design matrices and associated quadratic forms: structure and properties," CeMMAP working papers CWP16/04, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spacial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.

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