The Stein phenomenon for monotone incomplete multivariate normal data

  • Authors:

  • Donald St. P. Richards;Tomoya Yamada

  • Affiliations:

  • Department of Statistics, Penn State University, University Park, PA 16802, USA;Department of Economics, Sapporo Gakuin University, 11 Bunkyodai, Ebetsu, Hokkaido, Japan

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2010

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Abstract

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