Summary
LetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IRk, Δ ∈ IRk×k positive definite and symmetric andE(X−a)′ Δ−1(X−a)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale.
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Research supported by the Nuffield Foundation.
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Grübel, R. A minimal characterization of the covariance matrix. Metrika 35, 49–52 (1988). https://doi.org/10.1007/BF02613285
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DOI: https://doi.org/10.1007/BF02613285