Основы математической статистики. Лекция 8

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23:00 f(x, θ)=1/θ q(x/θ), log(f)=log(q(x/θ))−log(θ)
∂/∂θ log(f)=−x/θ^2∙q'(x/θ)/q(x/θ)−1/θ
I(θ)=∫ [∂/∂θ log(f)]^2 1/θ q(x/θ) dx=
=∫ [x/θ^2∙q'(x/θ)/q(x/θ)+1/θ]^2 1/θ q(x/θ) dx=
y=x/θ
=1/θ^2 ∫ [1+y∙q'(y)/q(y)]^2∙q(y)∙dy=1/θ^2 E( [1+y∙q'(y)/q(y)]^2 )

I(θ)=1/θ^2 ∙ E_1(Y^2), где Y=1+y∙log'(q(y)), E_1(Y^2)=∫ [1+y∙q'(y)/q(y)]^2∙q(y)∙dy