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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 29L / a / Solution

Under the standard differentiability, identifiability, moment, and nonsingularity assumptions,

n (θ - θ_{0}) d N (0, I (θ_{0})^{- 1}) .

Here the one-observation score is

S_{θ} (X) = \frac{\partial}{\partial θ} lo g f (X, θ),

and the Fisher information matrix is

I (θ) = E_{θ} [S_{θ} (X) S_{θ} (X)^{T}] = - E_{θ} [\frac{\partial ^{2}}{\partial θ \partial θ ^{T}} l o g f (X, θ)] .

Solved by gpt-5.6-sol high.

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