Quiz2 - 36-226 Summer 2010 Quiz 2 July 9 1 Indicate whether...

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Unformatted text preview: 36-226 Summer 2010 Quiz 2 July 9 1. Indicate whether the following are TRUE or FALSE. Write the entire word below each statement. Do not assume ANYTHING not written in the problem statement. You may use the result above. (a) The likelihood is always an increasing function of the unknown parameter θ. FALSE (b) Maximum likelihood estimators are minimum variance unbiased estimators. FALSE (ignore) (c) If θ is an unbiased estimator for θ and V ar(θ) → 0 as n → ∞, then θ is consistent for θ. TRUE (d) If θ is an unbiased estimator for θ and V ar(θ) → 0 as n → ∞, then θ is efficient for θ. FALSE (e) If θ is the MVUE for θ, then it has the smaller MSE than all other estimators for θ. FALSE (f) If θ is an estimator for θ but limn→∞ MSEθ (θ) = 0, then θ is not consistent for θ. FALSE 2. Let X1 , . . . , Xn be iid with density fX (x | β ) = 3β 3 I (x). x4 [β,∞) Show that X(1) is consistent for β . 3n 3 Useful results: E[X(1) ] = 3n−1 β and V ar[X(1) ] = (3n−1)2n n−2) β 2 . (3 We want to show that MSEβ (X(1) ) → 0 as n → ∞. First we find the bias bias(X(1) ) = E[X(1) ] − β = 3n 1 β−β = β. 3n − 1 3n − 1 Now, MSEβ (X(1) ) = bias(X(1) )2 + V ar[X(1) ] 3n 1 = β2 + β2 (3n − 1)2 (3n − 1)2 (3n − 2) (3n − 2)β 2 + 3nβ 2 = (3n − 1)2 (3n − 2) 2(3n − 1) = β2 (3n − 1)2 (3n − 2) 2 n→∞ = β2 − − 0 −→ (3n − 1)(3n − 2) Therefore, X(1) is consistent for β . 1 ...
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