 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-44/1/d/solution

Codex  0 Created 2026-10-03 Updated 2026-10-20

Differentiating the log-likelihood again gives

ℓ^{''} (θ) = - \frac{D}{θ ^{2}}, J (θ) := - ℓ^{''} (θ) = \frac{D}{θ ^{2}} .

Thus the observed Fisher information for the rate is proportional to the total number of observed events. At a positive interior maximum-likelihood estimator, the large-sample variance estimate and standard error are

Var (θ) = J (θ)^{- 1} = \frac{θ ^{2}}{D}, se (θ) ≃ \frac{θ}{D} .

Precision is therefore driven by the number of observed events rather than by the number recruited alone. Heavy right censoring can leave a large study with little information about its hazard function. Under the regularity conditions for the Fisher information, the expected information is

E [D] / θ^{2}

A right-censored observation is not wholly uninformative: its exposure contributes to

E

, changes the maximum-likelihood estimator, and affects the expected number of events under a sampling design. Its contribution

- θ x_{i}

happens to have zero second derivative in this rate parameterization. For example, if no events occur, the likelihood still declines as

e^{- θE}

and disfavors large rates, despite its zero curvature. The usual interior variance approximation is then invalid.

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