TY - JOUR

T1 - A paradox concerning shrinkage estimators

T2 - Should a known scale parameter be replaced by an estimated value in the shrinkage factor?

AU - Fourdrinier, Dominique

AU - Strawderman, William E.

N1 - Funding Information:
Received March 19, 1996. AMS 1991 subject classification: 62F10, 62H12, 62C20. Key words and phrases: spherical symmetry, quadratic loss, James Stein estimation, location parameter, minimaxity, robustness. * Research supported by National Science Foundation Grants DMS-9307727 and DMS-9400476.

PY - 1996/11

Y1 - 1996/11

N2 - When estimating, under quadratic loss, the location parameter θ of a spherically symmetric distribution with known scale parameter, we show that it may be that the common practice of utilizing the residual vector as an estimate of the variance is preferable to using the known value of the variance. In the context of Stein-like shrinkage estimators, we exhibit sufficient conditions on the spherical distributions for which this paradox occurs. In particular, we show that it occurs for t-distributions when the dimension of the residual vector is sufficiently large. The main tools in the development are upper and lower bounds on the risks of the James-Stein estimators which are exact at θ= 0.

AB - When estimating, under quadratic loss, the location parameter θ of a spherically symmetric distribution with known scale parameter, we show that it may be that the common practice of utilizing the residual vector as an estimate of the variance is preferable to using the known value of the variance. In the context of Stein-like shrinkage estimators, we exhibit sufficient conditions on the spherical distributions for which this paradox occurs. In particular, we show that it occurs for t-distributions when the dimension of the residual vector is sufficiently large. The main tools in the development are upper and lower bounds on the risks of the James-Stein estimators which are exact at θ= 0.

KW - James-stein estimation

KW - Location parameter

KW - Minimaxity

KW - Quadratic loss

KW - Robustness

KW - Spherical symmetry

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U2 - 10.1006/jmva.1996.0056

DO - 10.1006/jmva.1996.0056

M3 - Article

AN - SCOPUS:0030295571

SN - 0047-259X

VL - 59

SP - 109

EP - 140

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

IS - 2

ER -