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10_21ans - STAT 410 Examples for Fall 2009 3 a Let X 1 X 2...

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STAT 410 Examples for 10/21/2009 Fall 2009 3. Let X 1 , X 2 , … , X n be a random sample of size n from a uniform distribution on the interval ( 0 , θ ) . a) Obtain the method of moments estimator of θ , θ ~ . ( 29 2 θ X E = . 2 θ ~ X = . X 2 θ ~ = . b) Is θ ~ unbiased for θ ? That is, does E( θ ~ ) equal θ ? ( ( 29 2 θ X E X E = = . ( ( θ X 2 E θ ~ E = = . c) Obtain the maximum likelihood estimator of θ , θ ˆ . Likelihood function: ( 29 n n i θ 1 θ 1 θ L 1 = = = , θ > max X i , ( 0 θ L = , θ max X i . Therefore, θ ˆ = max X i .
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d) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? F max X i ( x ) = P ( max X i x ) = P ( X 1 x , X 2 x , … , X n x ) = P ( X 1 x ) P ( X 2 x ) P ( X n x ) = n x θ , 0 < x < θ . f max X i ( x ) = F ' max X i ( x ) = n n x n θ 1 - , 0 < x < θ . ( 29 θ 1 θ 0 θ 1 θ θ θ θ ˆ E 1 θ 0 θ 0 1 + = + = = = + - n n n x n dx x n dx x n x n n n n n n . θ ˆ is NOT unbiased for θ . e) What must c equal if c θ ˆ is to be an unbiased estimator for θ ? ( θ 1 θ 1 θ ˆ E 1 θ ˆ 1 E = + + = + = + n n n n n n n n . n n c 1 = . f) Compute Var( θ ~ ) and Var + θ ˆ 1 n n . X 2 θ ~ = . ( 29 ( 29 ( 29 n 2 σ 4 X Var 4 X 2 Var θ ~ Var = = = . For Uniform ( 0 , θ ) , 12 θ 2 2 σ = . ( 29 n = 3 θ θ ~ Var 2 . ( 29 2 θ 0 θ 2 θ θ θ θ ˆ E 2 2 θ 0 1 θ 0 1 2 2 + = + = = = + + - n n n x n dx x n dx x n x n n n n n n . ( 29 ( ( 29 [ ] ( 29 ( 29 2 2 2 2 2 2 1 2 θ 1 θ 2 θ θ ˆ E θ ˆ E θ ˆ Var + + = + - + = - = n n n n n n n . ( 29 ( 29 n n n n n n + = + = + 2 θ θ ˆ Var 1 θ ˆ 1 Var 2 2 .
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Def Let 1 θ ˆ and 2 θ ˆ be two unbiased estimators for θ .
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