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03_10ans - STAT 410 Examples for Spring 2008 4 a Let X 1 X...

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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 ( 0 , 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 = ± ² ³ ´ µ = = , θ > max X i , ( ) 0 ° L = , θ max X i . Therefore, ° ˆ = max X i .
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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 ° ± ² ³ ´ µ , 0 < x < θ . f max X i ( x ) = F ' max X i ( x ) = n n x n ° 1 - , 0 < x < θ . ( ) ° 1 ° 0 ° 1 ° ° ° ° ˆ E 1 ° 0 ° 0 1 + = ± ± ² ³ ´ ´ µ + = = = + - · · 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 = + + = + = ± ² ³ ´ µ + n n n n n n n n . n n c 1 + = . f) Compute Var( ° ~ ) and Var ± ± ² ³ ´ ´ µ + ° ˆ 1 n n . X 2 ° ~ = . ( ) ( ) ( ) n 2 ± 4 X Var 4 X 2 Var ° ~ Var = = = . For Uniform ( 0 , θ ) , 12 ° 2 2 ± = . ° ( ) n = 3 ° ° ~ Var 2 . ( ) 2 ° 0 ° 2 ° ° ° ° ˆ E 2 2 ° 0 1 ° 0 1 2 2 + = ± ± ² ³ ´ ´ µ + = = = + + - · · n n n x n dx x n dx x n x n n n n n n .
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