Asymptotic risk comparison of improved estimators for normal covariance matrix
The observed \(p\times p\) matrix \(S\) has a Wishart distribution with unknown scale matrix \(\Sigma\) and \(n\) degrees of freedom, with \(n>p+1\). If \(\hat\Sigma\) is an estimator of \(\Sigma\), the following two loss functions are considered: \[ L_1({\hat\Sigma},\Sigma) = \text{tr}({\hat\Sigma}\Sigma^{-1}) - \log|{\hat\Sigma}\Sigma^{- 1}| - p; \] \[ L_2({\hat\Sigma},\Sigma) = frac{1}{2}\text{tr}({\hat\Sigma}\Sigma^{-1} - I)^2. \] \textit{L. R. Haff} [Ann. Stat. 8, 586-597 (1980; Zbl 0441.62045)] showed that among the scalar multiples of \(S\), the best estimator under \(L_1\) is \(S/n\), and the best estimator under \(L_2\) is \(S/(n+p+1)\). Haff gave estimators improving on these scalar multiples of \(S\). In the present paper, the following new estimators are defined: for \(L_ 1\) the estimator is \((S+\gamma C)/n\) and for \(L_ 2\) the estimator is \((S+\gamma C)/(n+p+1)\), where C is a positive-definite matrix and \(\gamma\) is a positive value close to zero. A detailed numerical comparison of the risks (for large n) of these estimators and a minimax estimator for \(\Sigma\) given by \textit{W. James} and \textit{Ch. Stein} [Estimation with quadratic loss. Proc. 4th Berkeley Sympos. math. statist. Probab. 1, 361-379 (1961)] is carried out. For \(p\geq 6\), the James-Stein estimator dominates the estimators given by Haff. For each p the estimators proposed in the present paper are better than the James- Stein estimator for some \(\Sigma\).
scientific article; zbMATH DE number 3843002