(Solution) - Suppose that X1 Xn form a -(2025 Original AI-Free Solution)

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Suppose that X1, . . . , Xn form a random sample from an exponential distribution for which the value of the parameter ? is unknown (? > 0). Let ?(?) denote the prior p.d.f. of ?, and let
denote the Bayes estimator of ? with respect to the prior p.d.f. ?(?) when the squared error loss function is used. Let ? = ?2, and suppose that instead of estimating ?, it is desired to estimate the value of? subject to the following squared error loss function:
L(?, a) = (? ? a)2 for ? > 0 and a > 0.
Let ?? denote the Bayes estimator of ?. Explain why ? > ?2.