10_28 - STAT 410 p.m.f. or p.d.f. Examples for 10/28/2011 f...

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STAT 410 Examples for 10/28/2011 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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Example 1: Consider
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10_28 - STAT 410 p.m.f. or p.d.f. Examples for 10/28/2011 f...

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