Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions
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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.
KeywordsConvex cones Integration-by-parts Minimax estimate Multivariate normal mean Polyhedral cones Positively homogeneous set Quadratic loss Spherically symmetric distribution Weakly differentiable function
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- Bock M.E. (1982). Employing vague inequality information in the estimation of normal mean vectors (estimators that shrink to closed convex polyhedral). In: Gupta S.S., Berger J.O. (eds). Statistical decision theory and related topics III. (Vol. 1) Academic Press, New York, pp 169–193Google Scholar
- Johnstone I. (1988). On inadmissibility of some unbiased estimates of loss, Statistical Decision Theory and Related Topics 4 (eds. S.S. Gupta and J.O. Berger), 1, 361–379, Springer-Verlag, New York.Google Scholar
- Katz M.W. (1961). Admissible and minimax estimates of parameters in truncated spaces. The Annals of Mathematical Statistics 32:136–142Google Scholar
- Stoer J., Witzgall C. (1970). Convexity and optimization in finite dimensions I. Springer, Berlin Heidelberg, New YorkGoogle Scholar