AISM 53, 427-435

© 2001 ISM

## Information inequalities in a family of uniform distributions

### Masafumi Akahira^{1} and Kei Takeuchi^{2}

^{1}Institute of Mathematics,
University of Tsukuba, Ibaraki 305-8571, Japan

^{2}Faculty 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 $\e>0$ the average mean squared error
(MSE) of any estimator in the interval of $\t$ values of
length
$\e$ and centered at $\t_0$ 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.

**Source**
(TeX, DVI )