Improved loss estimation for a normal mean matrix (Q1755127)

For an \((n,m)\)-matrix \(X(\omega)\) composed of independent, normal distributed elements \(X_{ij}\) with mean \(M_{ij}\) and variance \(1\), consider estimates \(\hat{M}=\hat{M}(X)\) of the mean matrix \(M\), given an observation \(X\) of \(X(\omega)\). An estimator \(\hat{M}\) is evaluated by the estimation error \(L(X)=\|\hat{M}(X) - M\|^2\) with the Euclidean matrix norm \(\| \cdot \|^2\). The performance of a loss estimator \(\lambda(X)\) is defined by its mean square deviation \({E_M}(\lambda(X) - L(X))^2\) from \(L(X)\), given the mean matrix \(M\). Loss estimators are constructed which dominate the unique unbiased loss estimator (according to [\textit{C. Stein}, Proc. Prague Symp. Asympt. Stat., Vol. II, Prague 1973, 345--381 (1974; Zbl 0357.62020)]) for a large class of matrix mean estimators. For illustration of the results, numerical experiments are presented.