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# 10_07 - STAT 410 Examples for Fall 2011 Def An estimator θ...

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Unformatted text preview: STAT 410 Examples for 10/07/2011 Fall 2011 Def An estimator θ ˆ is said to be unbiased for θ if E( θ ˆ ) = θ for all θ . 3. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = - ≤ ≤ ⋅ otherwise 1 θ 1 θ θ 1 x x 0 < θ < ∞ . Recall: The method of moments estimator of θ is θ ~ = 1 X 1 X X 1- =- , the maximum likelihood estimator of θ is θ ˆ = ∑ = ⋅- n i i n 1 X ln 1 . d) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? ( ) ( ) ∫ ∫ = =- ∞ ∞- ⋅ ⋅ ⋅ 1 θ θ 1 X 1 θ 1 ln θ ln X ln E ; dx x x dx x f x . Integration by parts: ∫ ∫- = b a b a du v a b v u dv u Choice of u : L ogarithmic A lgebraic T rigonometric E xponential u = ln x , dv = dx x dx x 1 θ 1 θ θ 1 θ 1 θ 1-- ⋅ ⋅ = , du = dx x 1 , v = θ 1 x . ( ) ∫ ∫ - = = ⋅ ⋅ ⋅ ⋅- 1 θ 1 θ 1 1 θ θ 1 1 1 1 ln θ 1 ln X ln E dx x x x x dx x x = θ 1 θ 1 1 1 θ 1 1 1 θ 1 1 θ 1- = - =- = - ⋅ ⋅ ∫ ∫- x dx x dx x x . Therefore, ( ) ( ) ( ) ∑ ∑ = =-- =- = ⋅ ⋅ n i n i i n n 1 1 θ 1 X ln E 1 θ ˆ E = θ , that is, θ ˆ is an unbiased estimator for θ . OR F X ( x ) = x 1 / θ , 0 < x < 1....
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10_07 - STAT 410 Examples for Fall 2011 Def An estimator θ...

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