INFORMATION INEQUALITIES FOR THE BAYES RISK FOR A FAMILY OF NON-REGULAR DISTRIBUTIONS

MASAFUMI AKAHIRA AND NAO OHYAUCHI

Institute of Mathematics, University of Tsukuba, Ibaraki 305–8571, Japan

(Received April 20, 2001; revised September 20, 2001)

Abstract. For a family of non-regular distributions with a location parameter including the uniform and truncated distributions, the stochastic expansion of the Bayes estimator is given and the asymptotic lower bound for the Bayes risk is obtained and shown to be sharp. Some examples are also given.

Key words and phrases: Loss function, Bayes risk, Bayes estimator, lower bound, uniform distribution, truncated distributions.