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Asymptotic distributions of robust shape matrices and scales

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  • Frahm, Gabriel

Abstract

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix ∑ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for ∑ or another matrix Γ ∝ ∑. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the presented estimators are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the presented scale function leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine (2008) in the context of local asymptotic normality theory.

Suggested Citation

  • Frahm, Gabriel, 2008. "Asymptotic distributions of robust shape matrices and scales," Discussion Papers in Econometrics and Statistics 5/07, University of Cologne, Institute of Econometrics and Statistics.
    • Handle: RePEc:zbw:ucdpse:507
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    References listed on IDEAS

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    1. Salibian-Barrera, Matias & Van Aelst, Stefan & Willems, Gert, 2006. "Principal Components Analysis Based on Multivariate MM Estimators With Fast and Robust Bootstrap," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1198-1211, September.
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    More about this item

    Keywords

    local asymptotic normality; M-estimator; R-estimator; robust covariance matrix estimator; scale-invariant function; S-estimator; shape matrix; Tyler's M-estimator;
    All these keywords.

    JEL classification:

    • H20 - Public Economics - - Taxation, Subsidies, and Revenue - - - General
    • E20 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - General (includes Measurement and Data)

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