Stein's Initial Estimator of a Multivariate Normal Mean

• Activity Number: 513
• Type: Contributed
• Date/Time: Wednesday, August 3, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Jerome P. Keating*+ and Robert L. Mason
• Companies: The University of Texas at San Antonio and Southwest Research Institute
• Keywords: Stein rule ; Hyperbolic geometry ; Astrophysics

Abstract

In 1956, Charles Stein introduced the simple version of his famous Stein rule estimator, which has now been generalized in numerous ways. However, from the text of the original paper it is clear that Professor Stein first considered an alternative version, based on the positive-part of the square-root of the coefficient of his famous discovery. We investigate his initial-rule and its connection with hyperbolic geometry. The initial Stein-rule is shown to be a contraction toward the origin predicated on distance in hyperbolic space. The standard estimator of a multivariate normal mean is shown to be inadmissible to Stein's initial-rule as well. Stein's initial-rule has practical application in situations where the Stein rule appears to shrink the components of the estimated normal mean too much toward the origin (or any other specified point). Likewise, given its roots in hyperbolic geometry, Stein's initial-rule should have many applications in astrophysics as well.