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Asymptotic distribution of the conditional regret risk for selecting good exponential populations. (English) Zbl 1243.62006

Summary: Empirical Bayes methods are applied to construct selection rules for the selection of all good exponential distributions. We modify the selection rule introduced and studied by S S. Gupta and T. Liang [Sankhyā, Ser. B 61, No. 2, 289–304 (1999; Zbl 0972.62003)] who proved that the regret risk converges to zero with rate \(O(n^{-\lambda /2})\), \(0<\lambda \leq 2\). The aim of this paper is to study the asymptotic behavior of the conditional regret risk \(\mathcal {R}_{n}\). It is shown that \(n{\mathcal R}_{n}\) tends in distribution to a linear combination of independent \(\chi ^{2}\)-distributed random variables. As an application we give a large sample approximation for the probability that the conditional regret risk exceeds the Bayes risk by a given \(\varepsilon >0.\) This probability characterizes the information contained in the historical data.

MSC:

62C12 Empirical decision procedures; empirical Bayes procedures
62E20 Asymptotic distribution theory in statistics
62F07 Statistical ranking and selection procedures

Keywords:

Bayes method

Citations:

Zbl 0972.62003
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References:

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