Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series B, Pt. 2, pp. 289-304

SELECTING GOOD EXPONENTIAL POPULATIONS COMPARED WITH A CONTROL: A NONPARAMETRIC EMPIRICAL BAYES APPROACH

By

SHANTI S. GUPTA, * Purdue University, West Lafayette*

and

TACHEN LIANG, *Wayne State University, Detroit*

*SUMMARY.* This paper deals with empirical Bayes selection procedures for
selecting good exponential populations compared with a control.
Based on the accumulated historical data, an empirical Bayes
selection procedure} $\uw\delta^*_n$ {\footnotesize is constructed by mimicking the
behavior of a Bayes selection procedure. The empirical Bayes
selection procedure} $\uw\delta^*_n$ {\footnotesize is proved to be
asymptotically optimal. The analysis shows that the rate of convergence
of} $\uw \delta^*_n$ {\footnotesize is influenced by the tail probabilities of the underlying
distributions. It is shown that under certain regularity conditions on the
moments of the prior distribution, the empirical Bayes selection procedure}
$\uw \delta ^*_n$ {\footnotesize is asymptotically optimal of order $O(n^{- \lambda/2})$ for
some $0 < \lambda \leq 2$. A lower bound with rate of
convergence of order $n^{-1}$ is also established for the
regret Bayes risk of the empirical Bayes selection procedure}
$\uw\delta ^*_n$. {\footnotesize
This result suggests that a rate of order
$O(n^{-1})$ might be the best possible rate of convergence for
this empirical Bayes selection problem.

*AMS (1991) subject classification.*Primary 62F07; secondary 62C12.

*Key words and phrases. *Asymptotic optimality, comparison with
a control, empirical Bayes, good populations, rate of
convergence, regret Bayes risk, selection procedure.