STEIN CONFIDENCE SETS AND THE BOOTSTRAP

The Stein estimator ($) over cap xi(s) dominates the sample mean, under quadratic loss, in the N(xi, I) model of dimension q greater than or equal to 3. A Stein confidence set is a sphere of radius ($) over cap d centered at ($) over cap xi(s). The radius ($) over cap d is constructed to make the coverage probability converge to alpha as dimension q increases. This paper studies properties of Stein confidence sets for moderate to large values of 4. Our main results are: Stein confidence sets dominate the classical confidence spheres for xi under a geometrical risk criterion as q --> oo. Correct bootstrap critical values for Stein confidence sets require resampling from a N(($) over cap xi, I) distribution, where \($) over cap xi\ estimates \xi\ well. Simple asymptotic or bootstrap constructions of ($) over cap 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 greater than or equal to 5.