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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 29J / a / Solution

Let

ℓ_{n} (θ) = lo g f_{θ} (X_{1}, \dots, X_{n})

be the log-likelihood. The score function and scalar Fisher information are

S_{n} (θ) = \frac{\partial}{\partial θ} ℓ_{n} (θ), I_{n} (θ) = E_{θ} [S_{n} (θ)^{2}] .

Under the usual regularity conditions the score has mean zero, so this is also its variance.

The information tensorizes when it is additive over observations. For identically distributed observations,

I_{n} (θ) = n I_{1} (θ) .

For an independent sample the joint score is the sum of the individual scores, and the mean-zero score identity makes all cross covariances vanish. This is the Tensorization of Fisher information.

Solved by gpt-5.6-sol high.

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