Summary
The Bayesian estimation problem for the parameter θ of an exponential probability distribution is considered, when it is assumed that θ has a natural conjugate prior density and a loss-function depending on the squared error is used. It is shown that, with probability one, the posterior density of the Bayesian—centered and scaled parameter converges pointwise to the normal probability density. The weak convergence of the posterior distributions to the normal distribution follows directly. Both correct and incorrect models are studied and the asymptotic normality is stated respectively.
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Dumitrescu, M.E.B. On the normality a posteriori for exponential distributions, using the Bayesian estimation. Ann Inst Stat Math 39, 211–218 (1987). https://doi.org/10.1007/BF02491460
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DOI: https://doi.org/10.1007/BF02491460