 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/29l/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

θ_{n}

be the MLE and let

I (θ)

be the one-observation Fisher information matrix. One standard form of the Wald statistic is

W_{n} (θ) = n (θ_{n} - θ)^{T} I (θ_{n}) (θ_{n} - θ) .

Any consistent information estimator gives the same asymptotics. Under

H_{0}

W_{n} (θ_{0}) \Rightarrow χ_{p}^{2}

. If

c_{p, α}

is the

(1 - α)

quantile of

χ_{p}^{2}

, reject when

W_{n} (θ_{0}) > c_{p, α} .

The asymptotic type-I error is

α

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