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THE KULLBACK-LEIBLER RISK OF THE STEIN ESTIMATOR

AND THE CONDITIONAL MLE

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TAKEMI YANAGIMOTO

*The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu,
Minato-ku, Tokyo 106, Japan*
(Received October 29, 1992; revised June 22, 1993)

**Abstract.**
The decomposition of the Kullback-Leibler risk
of the maximum likelihood estimator (MLE) is discussed in relation to
the Stein estimator and the conditional MLE. A notable correspondence
between the decomposition in terms of the Stein estimator and that in
terms of the conditional MLE is observed. This decomposition reflects
that of the expected log-likelihood ratio. Accordingly, it is
concluded that these modified estimators reduce the risk by reducing
the expected log-likelihood ratio. The empirical Bayes method is
discussed from this point of view.

*Key words and phrases*:
Conditional inference, empirical
Bayes method, expected log-likelihood ratio, exponential dispersion
model, maximum likelihood estimator, Stein estimator.

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