 Solution

ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-1/a1/d/solution

Write

s = \sum_{i} lo g x_{i} < 0

ℓ^{'} (θ) = \frac{N}{θ + 1} + s .

Setting it to zero gives the maximum-likelihood estimator

θ = - \frac{N}{\sum _{i} lo g x _{i}} - 1 .

This lies above

- 1

J (θ) = - ℓ^{''} (θ) = \frac{N}{( θ + 1 ) ^{2}} > 0 .

Consequently the log-likelihood is strictly concave throughout its domain. Its stationary point is the unique global maximum, not merely a candidate local maximum. Indeed the log-likelihood tends to minus infinity as either

θ ↓ - 1

θ \to \infty

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