 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/29k/b/solution

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

θ \in R^{p}

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

Replacing

I (θ_{n})

I (θ_{0})

gives the same limit.

Under

P_{θ_{0}}

I (θ_{n})^{1/2} n (θ_{n} - θ_{0}) d N_{p} (0, I_{p}) .

Consequently

W_{n} (θ_{0}) d χ_{p}^{2},

using the characterization of the chi-squared distribution as the squared norm of a standard normal vector.

q_{p, 1 - α}

is the

(1 - α)

quantile of

χ_{p}^{2}

, an asymptotic

(1 - α)

confidence region is

C_{n} = {θ \in R^{p} : W_{n} (θ) \leq q_{p, 1 - α}} .

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