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LINEAR VERSUS NONLINEAR RULES FOR

MIXTURE NORMAL PRIORS

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BRANI VIDAKOVIC

*Institute of Statistics and Decision Sciences, Duke University,*

Box 90251, Durham, NC 27708-0251, U.S.A.
(Received March 10, 1997; revised September 29, 1997)

**Abstract.**
The problem under consideration is the
Gamma-minimax estimation, under *L*_{2} loss, of a
multivariate normal mean when the covariance matrix is known.
The family Gamma of priors is induced by mixing zero mean
multivariate normals with covariance matrix *tau I* by
nonnegative random variables *tau*, whose distributions
belong to a suitable family \cal G. For a fixed family
\cal G, the linear (affine) Gamma-minimax rule is
compared with the usual Gamma-minimax rule in terms of
corresponding Gamma-minimax risks. It is shown that the
linear rule is ``good'', i.e., the ratio of the risks is
close to 1, irrespective of the dimension of the model. We
also generalize the above model to the case of nonidentity
covariance matrices and show that independence of the
dimensionality is lost in this case. Several examples
illustrate the behavior of the linear Gamma-minimax rule.

*Key words and phrases*:
Affine rules, efficiency,
Bayes risk, Brown's identity, Gamma-minimax rules.

**Source**
( TeX ,
DVI ,
PS )