The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from Np+q(µ, Σ), a 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 bµ the maximum likelihood estimator of µ. We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than bµ under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators improve on their unrestricted counterparts. We derive results for the case in which Σ is block-diagonal, the loss function is quadratic and nonspherical, and the shrinkage estimator is constructed by means of a non-decreasing, differentiable function of a quadratic form in bµ. In the case of the problem of shrinking bµ to a vector whose components have a comm...

• Type: Journal
• Year: 2010
• Where: MA
• Authors: Donald St. P. Richards, Tomoya Yamada
• Keywords: Communications, Estimators, MA 2010, Shrinkage Estimators, Unrestricted Counterparts