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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

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.

Suggested Citation

  • Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(1), pages 1-28, February.
    • Handle: RePEc:cup:etheor:v:17:y:2001:i:01:p:1-28_17
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    Cited by:

    1. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.
    2. J. Roderick McCrorie, 2021. "Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 244-281, November.
    3. Patrick Marsh, 2019. "Properties of the power envelope for tests against both stationary and explosive alternatives: the effect of trends," Discussion Papers 19/03, University of Nottingham, Granger Centre for Time Series Econometrics.
    4. Hillier, Grant & Martellosio, Federico, 2006. "Spatial design matrices and associated quadratic forms: structure and properties," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 1-18, January.
    5. 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.
    6. Robinson, Peter M. & Rossi, Francesca, 2012. "Improved tests for spatial correlation," MPRA Paper 41835, University Library of Munich, Germany.
    7. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers 44/13, Institute for Fiscal Studies.
    8. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Aman Ullah & Yong Bao & Yun Wang, 2014. "Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process," Working Papers 201413, University of California at Riverside, Department of Economics.
    10. repec:cep:stiecm:/2013/565 is not listed on IDEAS
    11. Grant Hillier & Federico Martellosio, 2004. "Spatial design matrices and associated quadratic forms: structure and properties," CeMMAP working papers 16/04, Institute for Fiscal Studies.
    12. 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.
    13. Giovanni Forchini, "undated". "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.

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