TY - JOUR
T1 - The Stein phenomenon for monotone incomplete multivariate normal data
AU - Richards, Donald St P.
AU - Yamada, Tomoya
N1 - Funding Information:
The first author was supported in part by National Science Foundation grant DMS-0705210. The second author’s research was supported by a sabbatical leave-of-absence from Sapporo Gakuin University.
PY - 2010/3
Y1 - 2010/3
N2 - We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from Np + q (μ, Σ), a (p + q)-dimensional multivariate normal population with mean μ and covariance matrix Σ. On the basis of data consisting of n observations on all p + q characteristics and an additional N - n observations on the last q characteristics, where all observations are mutually independent, denote by over(μ, ̂) the maximum likelihood estimator of μ. We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than over(μ, ̂) under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which Σ is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in over(μ, ̂). For the problem of shrinking over(μ, ̂) to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.
AB - We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from Np + q (μ, Σ), a (p + q)-dimensional multivariate normal population with mean μ and covariance matrix Σ. On the basis of data consisting of n observations on all p + q characteristics and an additional N - n observations on the last q characteristics, where all observations are mutually independent, denote by over(μ, ̂) the maximum likelihood estimator of μ. We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than over(μ, ̂) under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which Σ is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in over(μ, ̂). For the problem of shrinking over(μ, ̂) to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.
UR - http://www.scopus.com/inward/record.url?scp=72549090105&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=72549090105&partnerID=8YFLogxK
U2 - 10.1016/j.jmva.2009.11.002
DO - 10.1016/j.jmva.2009.11.002
M3 - Article
AN - SCOPUS:72549090105
SN - 0047-259X
VL - 101
SP - 657
EP - 678
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
IS - 3
ER -