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Bhattacharyya bounds are considered for the unbiased estimation of a parametric function when the sampling distribution is a member of an exponential family of distributions. It is shown that the Bhattacharyya bounds converge to the variance of the best unbiased estimator. The application of this result to variance determination in demonstrated with examples from the negative binomial distribution and from the exponential distribution in a reliability theory context.
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Blight et al. (1974) studied this question.
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