The Stein phenomenon for monotone incomplete multivariate normal data

Donald St. P. Richards and Tomoya Yamada

Journal of Multivariate Analysis, 2010, vol. 101, issue 3, 657-678

Abstract: We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . 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 the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than 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 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 . For the problem of shrinking 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.

Keywords: Cauchy's; interlacing; theorem; Empirical; Bayes; estimation; James-Stein; estimator; Missing; completely; at; random; Positive-part; estimator; Shrinkage; estimator; Squared-error; loss; function; Wishart; distribution (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (2)

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