HW3 - ORIE6700 Homework 3 Fall 2010 1. Prove that if T (X )...

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Unformatted text preview: ORIE6700 Homework 3 Fall 2010 1. Prove that if T (X ) is a sufficient statistic for θ, then the posterior density for θ depends on x only through T (x). 2. Suppose S is an unbiased estimator of θ and b = 0. Why is S + b inadmissible? 3. C & B 7.6 (a)-(b) on p. 355 4. C & B 7.7 on p. 355 5. C & B 7.10 on p. 356 1 1 1 6. Let X1 , . . . , Xn , be a sample from U (θ − 2 , θ + 1 ), the uniform distribution on (θ − 2 , θ + 2 ). Find an 2 MLE of θ. Is the MLE unique? 7. Let X1 , . . . , Xn , n ≥ 2, be iid with common density f (x|θ) = 1 exp{−(x − µ)/σ }, σ x ≥ µ, where θ = (µ, σ 2 ), −∞ < µ < ∞, σ 2 > 0. (a) Find the MLE of µ and σ 2 . (b) Say we know that µ = 0; give the MLE for σ 2 . Then find the MLE of Pθ [X ∗ ≥ t] for t ≥ µ, where we still assume that µ = 0 and where X ∗ ∼ f (x|θ) is a hypothetical new observation (this is an example of prediction, where we use what we have learned from the first n observations to predict the distribution of the next observation) 8. C & B 7.33 on p. 361 ...
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This note was uploaded on 12/14/2010 for the course ORIE 6700 taught by Professor Woodard during the Fall '10 term at Cornell University (Engineering School).

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