Let c > 0 and consider the loss function Assume that has a continuous distribution. Prove

Let c > 0 and consider the loss function

Assume that θ has a continuous distribution. Prove that a Bayes estimator of θ will be any 1/(1+ c) quantile of the posterior distribution of θ. The proof is a lot like the proof of Theorem 4.5.3. The result holds even if θ does not have a continuous distribution, but the proof is more cumbersome.

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