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The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with Application to the AR(1) Model

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  • Giovanni Forchini

Abstract

Often neither the exact density nor the exact cumulative distribution function (CDF) of a statistic of interest are available in the statistics and econometrics literature (for example the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact CDF of a statistic of interest is very complicated despite the statistic being “simple” (for example the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the CDF of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the CDF of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact CDF of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialised to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

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  • Giovanni Forchini, "undated". "The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with Application to the AR(1) Model," Discussion Papers 01/02, Department of Economics, University of York.
  • Handle: RePEc:yor:yorken:01/02
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    Cited by:

    1. Robinson, Peter M. & Rossi, Francesca, 2012. "Improved tests for spatial correlation," MPRA Paper 41835, University Library of Munich, Germany.
    2. Patrick Marsh, "undated". "A Measure of Distance for the Unit Root Hypothesis," Discussion Papers 05/02, Department of Economics, University of York.
    3. 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.
    4. Vougas, Dimitrios V., 2006. "Remark on the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root autoregression," Statistics & Probability Letters, Elsevier, vol. 76(1), pages 27-34, January.
    5. 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.
    6. 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.
    7. repec:cep:stiecm:/2013/565 is not listed on IDEAS

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