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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

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