Statistica Sinica 5(1995), 109-127

STEIN CONFIDENCE SETS AND THE BOOTSTRAP

Rudolf Beran

University of California, Berkeley

Abstract: The Stein estimator dominates the sample mean, under quadratic loss, in the N(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 dimension q increases. This paper studies properties of Stein confidence sets for moderate to large values of q. Our main results are:
•Stein confidence sets dominate the classical confidence spheres for ξ under a geometrical risk criterion as q→ ∞.
•Correct bootstrap critical values for Stein confidence sets require resampling from a distribution, where estimates |ξ'| well.
•Simple asymptotic or bootstrap constructions of d result in a coverage probability error of O(q-1/2). A more sophisticated bootstrap approach reduces coverage probability error to O(q-1). The faster rate of convergence manifests itself numerically for q≥5.

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