08_28_09ans - STAT 409 5 Let X 1 X 2 X n be a random sample...

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STAT 409 Examples for 08/28/2009 Fall 2009 5. Let X 1 , X 2 , … , X n be a random sample of size n from a uniform distribution on the interval ( 0 , θ ) . f ( x ; θ ) = < < otherwise 0 0 1 θ θ x E ( X ) = 2 θ Var ( X ) = 12 2 θ F ( x ; θ ) = < < < θ θ θ 1 0 0 0 x x x x 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 ( 29 2 θ X E X E = = . ( 29 ( 29 θ X 2 E θ ~ E = = . c θ ~ is unbiased for θ . c) Compute Var( θ ~ ). 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 .
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d) 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 , ( 29 0 θ L = , θ < max X i . Therefore, θ ˆ = max X i . e) 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 .
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f) What must c equal if c θ ˆ is to be an unbiased estimator for θ ? ( 29
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

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08_28_09ans - STAT 409 5 Let X 1 X 2 X n be a random sample...

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