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.

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