Selecting a minimax estimator doing well at a point
Let \(X_ i\sim N(\theta_ i,I_ p)\), \(i=1,...,n\) \((n>p+1\), \(p>1)\) be an independent finite sequence of p-dimensional vectors. Let \(X=(X_ 1,...,X_ n)\) be a \(p\times n\) matrix of random variables with matrix of means \(\theta =(\theta_ 1,...,\theta_ n)\). Stein estimator is of the form \(\delta (X)=X+\nabla \ln f(\ell)\), where \(\ell =(\ell_ 1,...,\ell_ p)'\) is a vector with components \(\ell_ 1\geq \ell_ 2\geq...\geq \ell_ p>0\), the characteristic roots of \(S=XX'\), and \(\nabla \ln f(\ell)=((\partial /\partial X_{jk})\ln f(\ell))\) is the matrix of partial derivatives of the function ln f(\(\ell)\). In this paper the author gives an estimator \(\delta^*(X)\) which dominates \(\delta\) (X) such that the estimator is the best at \(\theta =0\) among estimators \(\delta_{\beta}(X)=\delta (X)-2X/(tr S/(p(p+1)-2)+\beta (\ell)\), where \(\beta\) is in a certain class.
scientific article; zbMATH DE number 3976110