Statistica Sinica

Rudolf Beran

Abstract:The Stein estimator dominates the sample mean, under quadratic loss, in theN(ξ,I) model of dimension q≥3. A Stein confidence set is a sphere of radius centered at . The radius is constructed to make the coverage probability converge to α as dimensionqincreases. This paper studies properties of Stein confidence sets for moderate to large values ofq. Our main results are:

•Stein confidence sets dominate the classical confidence spheres for ξ under a geometrical risk criterion asq→ ∞.

•Correct bootstrap critical values for Stein confidence sets require resampling from a distribution, where estimates |ξ'| well.

•Simple asymptotic or bootstrap constructions ofdresult in a coverage probability error ofO(q^{-1/2}). A more sophisticated bootstrap approach reduces coverage probability error toO(q^{-1}). The faster rate of convergence manifests itself numerically forq≥5.

Key words and phrases:Signal, white noise, coverage probability, geometrical risk.