Estimators Shrinking towards Projection Vector for Multivariate Normal Mean Vector under the Norm with a Known Interval

Abstract

Consider the problem of estimating a 􀆎 􀁚􀃎 mean vector 􀄾 (􀆎 􀃠 􀆐 􀀾 􀃐), 􀆐 􀃡 􀂙􀂈􀂕􀂒􀃞􀂤 􀃟 with a projection matrix 􀂤 under the quadratic loss, based on a sample 􀂲􀃎 􀃬 􀂲􀃏 􀃬 􀁺 􀃬 􀂲􀆌 . In this paper a James-Stein type estimator with shrinkage form is given when it’s variance distribution is specified and when the norm 􀃧􀃧 􀄾 􀃠 􀂤 􀄾 􀃧􀃧 is constrain, where 􀂤 is an idempotent and symmetric matrix and 􀂙􀂈􀂕􀂒􀃞􀂤􀃟 􀃡 􀆐 . It is characterized a minimal complete class of James-Stein type estimators in this case. And the subclass of James-Stein type estimators that dominate the sample mean is derived.