 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-4/13i/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

I (β) = ν \sum_{i} μ_{i}^{2} x_{i} x_{i}^{T} = ν (a b b c) .

Under standard interior, full-rank and large-sample regularity conditions, the maximum-likelihood estimator is asymptotically normal with mean

β

I (β)^{- 1} = \frac{1}{ν ( a c - b ^{2} )} (c - b - b a) .

Therefore

β_{2} \tilde{˙} N (β_{2}, \frac{a}{ν ( a c - b ^{2} )}) .

The sums already depend on sample size; no extra factor

1/ n

should be inserted. The variance is estimable by substituting fitted means. If all

z_{i}

are the same,

a c - b^{2} = 0

and the two coefficients are not separately identifiable.

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