Past exam of the statistics course of the University of Oxford / 2014 / 1 / a / i

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- Solution i

Solution

Write

I_{1} (θ)

for the Fisher information in one observation:

I_{1} (θ) = E_{θ} [{\frac{\partial}{\partial θ} lo g f (X; θ)}^{2}] = - E_{θ} [\frac{\partial ^{2}}{\partial θ ^{2}} lo g f (X; θ)] .

Under the stipulated regularity assumptions, the maximum-likelihood estimator is consistent and has asymptotic normality:

n (θ - θ) d N (0, I_{1} (θ)^{- 1}) .

Equivalently, its large-sample normal approximation has mean

θ

1/ [n I_{1} (θ)]

. The information for the whole random sample is

n I_{1} (θ)

; keeping the one-observation convention explicit prevents an extra factor of

n

.

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