TY - JOUR

T1 - Estimation of a location parameter with restrictions or "vague information" for spherically symmetric distributions

AU - Fourdrinier, Dominique

AU - Strawderman, William E.

AU - Wells, Martin T.

N1 - Funding Information:
Acknowledgements The authors express their gratitude to two referees who read the original manuscript with care and made a number of suggestions for improving the presentation. The support of NSF Grant DMS 97-04524 is gratefully acknowledged by W.E. Strawderman. The support of NSF Grant DMS 99-71586 is gratefully acknowledged by M.T. Wells

PY - 2006/3

Y1 - 2006/3

N2 - In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.

AB - In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.

KW - Convex cones

KW - Integration-by-parts

KW - Minimax estimate

KW - Multivariate normal mean

KW - Polyhedral cones

KW - Positively homogeneous set

KW - Quadratic loss

KW - Spherically symmetric distribution

KW - Weakly differentiable function

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U2 - 10.1007/s10463-005-0001-0

DO - 10.1007/s10463-005-0001-0

M3 - Article

AN - SCOPUS:33645700346

VL - 58

SP - 73

EP - 92

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

SN - 0020-3157

IS - 1

ER -