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The superharmonic condition for simultaneous estimation of means in exponential familles
Author(s) -
Haff L. R.,
Johnson R. W.
Publication year - 1986
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315035
Subject(s) - mathematics , estimator , subharmonic function , exponential function , bayes' theorem , statistics , james–stein estimator , exponential family , mathematical analysis , minimax estimator , bayesian probability , minimum variance unbiased estimator
The mean vector associated with several independent variates from the exponential subclass of Hudson (1978) is estimated under weighted squared error loss. In particular, the formal Bayes and “Stein‐like” estimators of the mean vector are given. Conditions are also given under which these estimators dominate any of the “natural estimators”. Our conditions for dominance are motivated by a result of Stein (1981), who treated the N p (θ, I ) case with p ≥ 3. Stein showed that formal Bayes estimators dominate the usual estimator if the marginal density of the data is superharmonic. Our present exponential class generalization entails an elliptic differential inequality in some natural variables. Actually, we assume that each component of the data vector has a probability density function which satisfies a certain differential equation. While the densities of Hudson (1978) are particular solutions of this equation, other solutions are not of the exponential class if certain parameters are unknown. Our approach allows for the possibility of extending the parametric Stein‐theory to useful nonexponential cases, but the problem of nuisance parameters is not treated here.