INFORMATION INEQUALITIES IN A FAMILY OF UNIFORM DISTRIBUTIONS

MASAFUMI AKAHIRA^{1} AND KEI TAKEUCHI^{2}

1Institute of Mathematics, University of Tsukuba, Ibaraki 305–8571, Japan 2Faculty of International Studies, Meiji-Gakuin University, Kamikurata-cho 1598, Totsuka-ku, Yokohama 244–0816, Japan

(Received June 2, 1999; revised January 21, 2000)

Abstract. For a family of uniform distributions, it is shown that for any small ε > 0 the average mean squared error (MSE) of any estimator in the interval of θ values of length ε and centered at θ_{0} can not be smaller than that of the midrange up to the order o(n^{−2}) as the size n of sample tends to infinity. The asymptotic lower bound for the average MSE is also shown to be sharp.

Key words and phrases: Best location equivariant estimator, average mean squared error, sufficient statistic.