THE KULLBACK-LEIBLER RISK OF THE STEIN ESTIMATOR AND THE CONDITIONAL MLE

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.