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Inferential Statisticshard
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Given a sample of nnn independent observations from a distribution f(x∣θ)f(x|\theta)f(x∣θ), why does the 'Observed' Fisher Information J(θ)=−∂2∂θ2ℓ(θ)J(\theta) = -\frac{\partial^2}{\partial \theta^2} \ell(\theta)J(θ)=−∂θ2∂2​ℓ(θ) often replace the 'Expected' Fisher Information I(θ)=E[J(θ)]I(\theta) = E[J(\theta)]I(θ)=E[J(θ)] in the calculation of confidence intervals?