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# 08_28_09 - STAT 409 5 Fall 2009 Examples for Let X 1 X 2 X...

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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 ~ . b) Is θ ~ unbiased for θ ? That is, does E( θ ~ ) equal θ ? c) Compute Var( θ ~ ). d) Obtain the maximum likelihood estimator of θ , θ ˆ . e) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? f) What must c equal if c θ ˆ is to be an unbiased estimator for θ ? g) Compute Var( θ ˆ ) and Var θ ˆ 1 n n . Def Let 1 θ ˆ and 2 θ ˆ be two unbiased estimators for θ . 1 θ ˆ is said to be more efficient than 2 θ ˆ if Var( 1 θ ˆ ) < Var( 2 θ ˆ ). The relative efficiency of 1 θ ˆ with respect to 2 θ ˆ is Var( 2 θ ˆ ) / Var( 1 θ ˆ ). h) Which estimator for θ is more efficient, θ ~ or θ ˆ 1 n n ? What is the relative efficiency of θ ˆ 1 n n with respect to θ ~ ?

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