post11 - Example: X1, , Xn Poisson() f (x1, , xn; ) =...

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Fisher Information X 1 , ··· ,X n f ( x ; θ ) Ω , Ω is an interval I n ( θ )= E ± ∂θ ln f ( X 1 , n ; θ ) ² 2 ! I ( θ E ± ln f ( X ; θ ) ² 2 ! 1 Example : X 1 , n Poisson( θ ) f ( x 1 , ,x n ; θ ln f ( x 1 , n ; θ ln f ( x 1 , n ; θ ± ln f ( x 1 , n ; θ ) ² 2 = E ± ln f ( x 1 , n ; θ ) ² 2 = I n ( θ 2 Assumptions: (regular cases) i) we can differentiate under the integral or summation sign, i.e. ( Z dx Z ( ) dx ii) θ does not appear in endpoints of the in- terval in which f ( x ; θ ) > 0. Note that exponential family pdf’s meet these conditions. If the support of a pdf depends on θ , it will not satisfy the conditions. With regular cases, (i) E ³ ln f ( X ; θ ) ´ =0 (ii) I ( θ - E ± 2 2 ln f ( X ; θ ) ² 3 Fisher Information for iid samples I n ( θ E X i ln f ( X i ; θ ) 2 = X i E ± ln f ( X i ; θ ) ² 2 ! +2 X i<j E ± ln f ( X i ; θ ) ln f ( X j ; θ ) ² = X i E ± ln f ( X i ; θ ) ² 2 ! = nI ( θ ) 4
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Example : X 1 , ··· ,X n N ( θ, σ 2 ), σ 2 is known f ( x 1 , ,x n ; θ )= ln f ( x 1 , n ; θ ∂θ ln f ( x 1 , n ; θ
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue University-West Lafayette.

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post11 - Example: X1, , Xn Poisson() f (x1, , xn; ) =...

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