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PREVIOUS EXAM I – STAT 543 1. Let X1 , . . . , Xn be a random sample from a GAMMA(α, θ) distribution, θ > 0, where α is a known constant. Assume that the Cramér-Rao regularity conditions are satisfied. (a) Find the Cramér-Rao lower bound for estimating θ. (b) Find the UMVUE of θ. Justify your answer. 2. Let X1 , X2 , . . . , be a sequence of iid random variables with common pdf ½ 2 −3 3x θ if 0 < x ≤ θ f (x|θ) = 0 otherwise, θ > 0. Also let γ(θ) ≡ log(1 + θ2 ). (a) Find the Method of Moments estimator Tn of γ(θ) based on X1 , . . . , Xn . (b) Show that {Tn } is consistent for γ(θ). Clearly state any standard results that you are using. (c) Show that the maximum likelihood estimator θ̂n of θ based on X1 , . . . , Xn is given by θ̂n = max{X1 , . . . , Xn }. 2 if y ≤ 0 0 3n (d) Show that Pθ (θ̂n ≤ y) = (y/θ) if 0 < y ≤ θ 1 if y ≥ θ (e) Using part(d) or otherwise, show that {θ̂n } is consistent for θ. 3 3. Let X1 , . . . , Xn be a random sample from N (2, 1/θ), θ > 0. We want to find the Bayes estimator of θ w.r.t. the prior GAMMA(α + 32 , 1), where α > 0 is a known constant. (a) Find the posterior distribution of θ. (b) Find the Bayes estimator under the loss function L(t, θ) = (t−θ)2 /θ2 . Justify your answer. 4