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Stein Estimation in High Dimensions and the Bootstrap

Beran, Rudolf

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Abstract

The Stein estimator and the better positive-part Stein estimatorboth dominate the sample mean, under quadratic loss, in the standardmultivariate model of dimension q. Standard large sample theory does notexplain this phenomenon well. Plausible bootstrap estimators for the riskof the Stein estimator do not converge correctly at the shrinkage point assample size n increases. By analyzing a submodel exactly, with the helpof results from directional statistics, and then letting dimension q go toinfinity, we find:a) In high dimensions, the Stein and positive-part Stein estimators areapproximately admissible and approximately minimax on large compact ballsabout the shrinkage point. The sample mean is neither.b) A new estimator, asymptotically equivalent as dimension q tends toinfinity, appears to dominate the positive-part Stein estimator slightlyfor finite q.c) Resampling from a fitted standard multivariate normal distribution inwhich the length of the fitted mean vector estimates the length of thetrue mean vector well is the key to consistent bootstrap risk estimationfor Stein estimators.

Item Type: Working paper
Place of Publication: Heidelberg
Edition: revised August 1993
Date Deposited: 27 Jun 2016 15:06
Date: August 1993
Number of Pages: 19
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
Uncontrolled Keywords: Admissible; minimax; high dimension; orthogonal group; equivariant; directional statistics
Schriftenreihe ID: Beiträge zur Statistik > Beiträge
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