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

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  • Hillier, G.H.

Abstract

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

Suggested Citation

  • Hillier, G.H., 1999. "The density of a quadratic form in a vector uniformly distributed on the n-sphere," Discussion Paper Series In Economics And Econometrics 9902, Economics Division, School of Social Sciences, University of Southampton.
  • Handle: RePEc:stn:sotoec:9902
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    Cited by:

    1. Giovanni Forchini & Patrick Marsh, "undated". "Exact Inference for the Unit Root Hypothesis," Discussion Papers 00/54, Department of Economics, University of York.
    2. Robinson, Peter M. & Rossi, Francesca, 2012. "Improved tests for spatial correlation," MPRA Paper 41835, University Library of Munich, Germany.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(01), pages 211-242, February.
    8. 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.
    9. repec:cep:stiecm:/2013/565 is not listed on IDEAS
    10. 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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