03_10 - STAT 410 Examples for 03/10/2008 Spring 2008 4. Let...

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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 ) . f ( x ) = ± ² ³ < < otherwise 0 0 1 x E ( X ) = θ Var ( X ) = 12 2 F ( x ) = ± ² ³ < < 1 0 0 0 x x x x a) Obtain the method of moments estimator of θ , ~ . b) Is ~ unbiased for θ ? That is, does E( ~ ) equal θ ? c) Obtain the maximum likelihood estimator of θ , ˆ . d) Is ˆ unbiased for θ ? That is, does E( ˆ ) equal θ ? e) What must c equal if c ˆ is to be an unbiased estimator for θ ? f) Compute Var( ~ ) and Var ´ ´ µ · · ¸ ¹ + ˆ 1 n n . Def Let 1 ˆ and 2 ˆ be two unbiased estimators for θ . 1 ˆ is said to be more efficient than 2 ˆ if Var( 1 ˆ ) < Var( 2 ˆ ). The relative efficiency of 1 ˆ with respect to 2 ˆ is Var( 2 ˆ ) / Var( 1 ˆ ). g)
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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_10 - STAT 410 Examples for 03/10/2008 Spring 2008 4. Let...

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