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Estimating normal tail probabilities. (English) Zbl 0601.62036

This paper considers the MLE, UMVUE and Bayes estimators of the normal distribution function \(\Phi\) [(a-\(\mu)\)/\(\sigma\) ] with priors \(\lambda (\mu /\sigma)\sigma^{-\alpha}\). In particular for \(\lambda (\eta)=\exp \{-\eta^ 2/2\tau^ 2\}\) it is shown that the best Bayes estimator w.r.t. squared error loss, corresponds to \(\tau =\infty\). However, for large n and \(\eta\to \infty\) the Bayes risk of the optimum Bayes estimator is larger than that of MLE which in itself is larger than the risk function of the UMVUE. This indicates that in the presence of nuisance parameters, in multi-parameter exponential family, a generalized Bayes estimator could be inadmissible.
Reviewer: B.K.Kale

MSC:

62F10 Point estimation
62F15 Bayesian inference
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