Suppose that X1, . . . , Xn form a random sample from an exponential distribution for which the value of the parameter θ is unknown (θ > 0). Let ξ(θ) denote the prior p.d.f. of θ, and let
denote the Bayes estimator of θ with respect to the prior p.d.f. ξ(θ) when the squared error loss function is used. Let ψ = θ2, and suppose that instead of estimating θ, it is desired to estimate the value ofψ subject to the following squared error loss function:
L(ψ, a) = (ψ − a)2 for ψ > 0 and a > 0.
Let ˆψ denote the Bayes estimator of ψ. Explain why ψ > θ2.