 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/o/o1/a/solution

Let

ℓ_{n} (θ) = \sum_{i = 1}^{n} lo g f (X_{i}; θ)

be the log-likelihood. Under the usual regularity conditions, the expected Fisher information in the whole sample is

I_{n} (θ) = E_{θ} [{ℓ_{n}^{'} (θ)}^{2}] = - E_{θ} [ℓ_{n}^{''} (θ)] = n I_{1} (θ) .

The equality of the two expressions uses mean-zero score and permitted differentiation of the normalization identity. Independent observations make the information additive. The maximum-likelihood estimator has asymptotic normality:

n (θ - θ) d N (0, I_{1} (θ)^{- 1}), θ approximately N (θ, I_{n} (θ)^{- 1}) .

The factor

n

is included in

I_{n}

; it must not be inserted a second time into the variance.

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