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EMPIRICAL BAYES WITH RATES AND BEST RATES

OF CONVERGENCE IN *u*(*x*)*C*(*theta*)exp(-*x*/*theta*)-FAMILY:

ESTIMATION CASE

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R. S. SINGH^{1} AND LAISHENG WEI^{2}

^{1} *Department of Mathematics and Statistics, University of Guelph,*

Guelph, Ontario, Canada N1G 2W1

^{2} *Department of Mathematics, University of Science and Technology of China,*

Hefei, Anhui, P. R. China
(Received May 30, 1990; revised June 24, 1991)

**Abstract.**
Let {(*X*_{i},*theta*_{i})} be a sequence of independent
random
vectors where *X*_{i}, conditional on *theta*_{i}, has the probability density
of the
form *f*(*x* | *theta*_{i}) = *u*(*x*)*C*(*theta*_{i})exp(-*x*/*theta*_{i}) and the unobservable
*theta*_{i} are i.i.d. according to an unknown *G* in some class \cal G of prior
distributions on *Theta*, a subset of {*theta* > 0 | *C*(*theta*) = \big( \int
*u*(*x*)exp(-*x*/*theta*)*dx*\big)^{-1} > 0}. For a
\cal S(*X*_{1},....,*X*_{n},*X*_{n+1})-measurable
function *phi*_{n}, let *R*_{n} = *E*(*phi*_{n}-*theta*_{n+1})^{2} denote the Bayes risk of
*phi*_{n} and let *R*(*G*) denote the infimum Bayes risk with respect to *G*.
For each
integer *s* > 1 we exhibit a class of \cal S(*X*_{1},....,*X*_{n},*X*_{n+1})-measurable
functions *phi*_{n} such that for *delta* in [*s*^{-1},1], *c*_{0}*n*^{-2s/(1+2s)} __<__
*R*_{n}(*phi*_{n},*G*)-*R*(*G*)__<__ *c*_{1}*n*^{-2(sdelta-1)/(1+2s)} under certain conditions
on *u*
and *G*. No assumptions on the form or smoothness of *u* is made, however.
Examples
of functions *u*, including one with infinitely many discontinuities, are
given for
which our conditions reduce to some moment conditions on *G*. When *Theta* is
bounded, for each integer *s* > 1 \cal S(*X*_{1},....,*X*_{n},*X*_{n+1})-measurable
functions
*phi*_{n} are exhibited such that for *delta* in [2/*s*,1],
*c*'_{0}*n*^{-2s/(1+2s)} __<__
*R*(*phi*_{n},*G*)-*R*(*G*) __<__ *c*'_{1}*n*^{-2sdelta/(1+2s)}. Examples of functions *u*
and class
\cal G are given where the above lower and upper bounds are achieved.

*Key words and phrases*:
Exponential family, empirical Bayes,
estimation, asymptotic optimality, rates and best rates.

**Source**
( TeX ,
DVI ,
PS )