Canadian Journal of Statistics

Improved minimax estimation of a normal precision matrix

Journal Article

Abstract

Let Sp × p have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ−1 under the loss functions L1(σ) tr (σ) ‐ log |σ| and L2(σ) = tr (σ). James‐Stein‐type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte‐Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James‐Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.

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