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Inadmissibility of the best equivariant estimators of the variance- covariance matrix, the precision matrix, and the generalized variance under entropy loss. (English) Zbl 0634.62050
Let \(X=[X_ 1,...,X_ k]\) (p\(\times k)\) have independent columns, with \(X_ i\sim N(\xi_ i,\Sigma)\) and the \(\xi_ i\) and \(\Sigma\) (p\(\times p\) nonsingular) are unknown. Let S (p\(\times p)\sim W(n,\Sigma)\) be a Wishart matrix independent of X. Estimation of \(\Sigma\) by \({\hat \Sigma}\) is studied under two loss functions: \[ L_ 1({\hat \Sigma},\Sigma)=tr({\hat \Sigma}\Sigma^{-1})-\log | {\hat \Sigma}\Sigma^{-1}| -p,\quad and\quad L_ 2({\hat \Sigma},\Sigma)=tr(\Sigma {\hat \Sigma}^{-1})-\log | \Sigma {\hat \Sigma}^{-1}| -p. \] Under \(L_ 1\) the best equivariant estimator is \(n^{-1}S\). An improved estimator (called “testimator”, based on an idea of C. Stein) is presented, defined by: \({\hat \Sigma}=n^{-1}S\) unless Roy’s maximum root test accepts the hypothesis that all \(\xi_ i\) are 0 in which case \({\hat \Sigma}=(n+k)^{-1}(S+XX')\). A similar testimator is found under \(L_ 2.\)
The results are used for simultaneous estimation of \(\mu\) nd \(\Sigma\) in a sample from a p-variate N(\(\mu\),\(\Sigma)\) population. Estimation of \(| \Sigma |\) is handled in a similar way. Monte Carlo results on the improvement of the estimators are presented. Some results on sequential estimation are also given.
Reviewer: R.A.Wijsman

62H12 Estimation in multivariate analysis
62C15 Admissibility in statistical decision theory
62F10 Point estimation
62L12 Sequential estimation