09_23_11_1 - STAT 409 p.m.f or p.d.f Examples for f x Fall...

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STAT 409 Examples for 09/23/2011 Fall 2011 p.m.f. or p.d.f. f ( x ; θ ), θ Ω . θ θ ' f ( x ; θ ) f ( x ; θ ' ) f ( x ; θ ) have common support for all θ f ( x ; θ ) is twice differentiable as a function of θ f ( x ; θ ) dx can be twice differentiable under the integral sign as a function of θ θ ln f ( x ; θ ) is called the score function E [ θ ln f ( X ; θ ) ] = 0 Fisher Information : I ( θ ) = Var [ θ ln f ( X ; θ ) ] = E [ ( θ ln f ( X ; θ ) ) 2 ] = – E [ 2 2 θ ln f ( X ; θ ) ] . Rao-Cramer Lower Bound : X 1 , X 2 , … , X n i.i.d. f ( x ; θ ) Y = u ( X 1 , X 2 , … , X n ) E ( Y ) = k ( θ ) Var ( Y ) ( ) ( ) ( ) θ I θ 2 n k ' If E ( θ ˆ ) = θ , then Var ( θ ˆ ) ( ) θ I 1 n . Let θ ˆ be an unbiased estimator of θ . θ ˆ is called an efficient estimator of θ if and only if the variance of θ ˆ attains the Rao-Cramer lower bound.
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| 3 3 θ ln f ( x ; θ ) | < M ( x ), E ( M ( X ) ) < θ ˆ maximum likelihood estimator. For large n , θ ˆ is approximately ( ) θ I θ 1 , N n . ± θ ˆ ( ) θ ˆ I 1 α 2 n z would have an approximate 100 ( 1 – α ) % confidence level for large n . Example 1: Let X be a Poisson ( λ ) random variable. That is, f ( x ; λ ) = ! λ λ x e x - , x = 0, 1, 2, 3, … . ln f ( x ; λ ) = ! λ λ ln ln x x - - λ ln f ( x ; λ ) = 1 λ - x 2 2 λ ln f ( x ; λ ) = 2 λ x - I ( λ ) = – E [ 2 2 λ ln f ( X ; λ ) ] = – E [ 2 λ X - ] = 2 λ 1 E ( X ) = 2 λ 1 λ = λ 1 OR I ( λ ) = Var [ λ ln f ( X ; λ ) ] = Var [ 1 X λ - ] = 2 λ 1 Var
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This note was uploaded on 10/13/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.

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09_23_11_1 - STAT 409 p.m.f or p.d.f Examples for f x Fall...

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