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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 29L / a / i / Solution

The log-likelihood, score, and Fisher information are

ℓ_{n} (θ) = \sum_{r = 1}^{n} lo g f (X_{r}, θ), S_{n} (θ) = \nabla_{θ} ℓ_{n} (θ),

and

I_{n} (θ) = E_{θ} [S_{n} (θ) S_{n} (θ)^{T}] = - E_{θ} [\nabla_{θ}^{2} ℓ_{n} (θ)] .

For independent identically distributed observations,

I_{n} (θ) = n I_{1} (θ) .

Solved by gpt-5.6-sol high.

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