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EMPIRICAL BAYES SEQUENTIAL ESTIMATION FOR

EXPONENTIAL FAMILIES: THE UNTRUNCATED COMPONENT

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ROHANA J. KARUNAMUNI

*Department of Mathematical Sciences, University of Alberta,*

Edmonton, Alberta, Canada T6G 2G1
(Received July 20, 1995; revised January 19, 1996)

**Abstract.**
We consider the empirical Bayes decision
problem where the component problem is the sequential estimation of
the mean *theta* of one-parameter exponential family of
distributions with squared error loss for the estimation error and a
cost *c* > 0 for each observation. The present paper studies the
*untruncated* sequential component case. In particular, an
untruncated asymptotically pointwise optimal sequential procedure is
employed as the component. With sequential components, an empirical
Bayes decision procedure selects both a stopping time and a terminal
decision rule for use in the component with parameter *theta*. The
goodness of the empirical Bayes sequential procedure is measured by
comparing the asymptotic behavior of its Bayes risk with that of the
component procedure as the number of past data increases to infinity.
Asymptotic risk equivalence of the proposed empirical Bayes
sequential procedure to the component procedure is demonstrated.

*Key words and phrases*:
Empirical Bayes estimation,
sequential components, asymptotically pointwise optimal,
asymptotically optimal.

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