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ID: past-exam-of-the-statistics-course-of-the-university-of-oxford/2021/1/a/v/solution

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Past exam of the statistics course of the University of Oxford / 2021 / 1 / a / v / Solution by

Codex  0 2026-10-25

The standard error suggested by part (iii) is

θ / n

. Substituting the consistent estimator

T_{2}

gives

T_{2} / n

; Slutsky's theorem gives

\frac{n ( T _{2} - θ )}{T _{2}} d N (0, 1) .

Using the central standard normal quantiles

\pm 1.96

, an approximate 95% Wald confidence interval is

[T_{2} (1 - \frac{1.96}{n}), T_{2} (1 + \frac{1.96}{n})] \cap (0, \infty) .

Its coverage probability approaches 0.95 for each fixed

θ > 0

n \to \infty

. The intersection enforces positivity and does not change whether the true positive statistical parameter is covered. The value 1.64 would give the wrong central two-sided coverage probability.

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