Non-minimaxity of debiased shrinkage estimators

Abstract

We consider the estimation of the p-variate normal mean of X∼Np(θ,I)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\sim \\mathcal {N}_p(\ heta ,I)$$\\end{document} under the quadratic loss function. We investigate the decision theoretic properties of debiased shrinkage estimator, the estimator which shrinks towards the origin for smaller ‖x‖2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert x\\Vert ^2$$\\end{document} and which is exactly equal to the unbiased estimator X for larger ‖x‖2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert x\\Vert ^2$$\\end{document}. Such debiased shrinkage estimator seems superior to the unbiased estimator X, which implies minimaxity. However, we show that it is not minimax under mild conditions.