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Saddlepoint Approximations For Noncentral Quadratic Forms

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  • Marsh, Patrick W.N.

Abstract

Many estimators and tests are of the form of a ratio of quadratic forms in normal variables. Excepting a few very special cases little is known about the density or distribution of these ratios, particularly if we allow for noncentrality in the quadratic forms. This paper assumes this generality and derives saddlepoint approximations for this class of statistics. We first derive and prove the existence of an exact inversion based on the joint characteristic function. Then the saddlepoint algorithm is applied and the leading term found, and analytic justification of the asymptotic nature of the approximation is given. As an illustration we consider the calculation of sizes and powers of F-tests, where a new exact result is found.

Suggested Citation

  • Marsh, Patrick W.N., 1998. "Saddlepoint Approximations For Noncentral Quadratic Forms," Econometric Theory, Cambridge University Press, vol. 14(5), pages 539-559, October.
    • Handle: RePEc:cup:etheor:v:14:y:1998:i:05:p:539-559_14
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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. Broda, S. & Paolella, M.S., 2009. "Evaluating the density of ratios of noncentral quadratic forms in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1264-1270, February.
    3. Patrick Marsh, "undated". "Saddlepoint Approximations for Optimal Unit Root Tests," Discussion Papers 09/31, Department of Economics, University of York.
    4. Patrick Marsh, "undated". "Saddlepoint Approximations in Non-Stationary Time Series," Discussion Papers 00/57, Department of Economics, University of York.
    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.

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