03_10ans - STAT 410 Examples for 03/10/2008 Spring 2008 4....

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Unformatted text preview: STAT 410 Examples for 03/10/2008 Spring 2008 4. Let X 1 , X 2 , … , X n be a random sample of size n from a uniform distribution on the interval ( , θ ) . a) Obtain the method of moments estimator of θ , & ~ . ( ) 2 & X E = . & 2 & ~ X = . & X 2 & ~ = . b) Is & ~ unbiased for θ ? That is, does E( & ~ ) equal θ ? ( ) ( ) 2 & X E X E = = . & ( ) ( ) & X 2 E & ~ E = = . c) Obtain the maximum likelihood estimator of θ , & ˆ . Likelihood function: ( ) n n i & 1 & 1 & L 1 = ¡ ¢ £ ¤ ¥ ¦ = ∏ = , θ > max X i , ( ) & L = , θ ≤ max X i . Therefore, & ˆ = max X i . d) Is & ˆ unbiased for θ ? That is, does E( & 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 & & ¡ ¢ £ ¤ ¥ , 0 < x < θ . f max X i ( x ) = F ' max X i ( x ) = n n x n & 1- ⋅ , 0 < x < θ . ( ) & 1 & & 1 & & & & ˆ E 1 & & 1 ≠ + = & & ¡ ¢ £ £ ¤ ¥ + = = = ⋅ ⋅ ⋅ ⋅ ⋅ +- ¦ ¦ n n n x n dx x n dx x n x n n n n n n . & ˆ is NOT unbiased for θ . e) What must c equal if c & ˆ is to be an unbiased estimator for θ ? ( ) & 1 & 1 & ˆ E 1 & ˆ 1 E = + + = + = & ¡ ¢ £ ¤ ¥ + ⋅ ⋅ ⋅ ⋅ n n n n n n n n . n n c 1 + = . f) Compute Var( & ~ ) and Var & & ¡ ¢ £ £ ¤ ¥ + & ˆ 1 n n . X 2 & ~ = . ( ) ( ) ( ) n 2 ¡ 4 X Var 4 X 2 Var & ~ Var ⋅ = = = ....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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03_10ans - STAT 410 Examples for 03/10/2008 Spring 2008 4....

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