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 N_{p + 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.