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# 03_10 - STAT 410 Examples for Spring 2008 4 Let X 1 X 2 X n...

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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 ) . 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) Which estimator for θ is more efficient, ° ~ or ° ˆ 1 n n + ? What is the relative efficiency of ° ˆ 1 n n + with respect to ° ~ ?

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Chebyshev’s Inequality: Let X
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03_10 - STAT 410 Examples for Spring 2008 4 Let X 1 X 2 X n...

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