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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/28j/b/iv/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 28J / b / iv / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

The assumed maximum-likelihood estimator is the sample mean of

h (X_{i})

. Its central limit theorem is

n (θ_{MLE} - θ_{0}) d N (0, Var_{θ_{0}} (h (X))) .

For a regular model, the regular maximum-likelihood limit is

n (θ_{MLE} - θ_{0}) d N (0, I (θ_{0})^{- 1}) .

The two asymptotic variances must agree, so

I (θ_{0}) = \frac{1}{Var _{θ_{0}} ( h ( X ))} .

Consequently

Var (θ_{MLE}) = \frac{Var ( h ( X ))}{n} = \frac{1}{n I ( θ _{0} )}

to first asymptotic order. The estimator therefore attains the Cramer-Rao bound asymptotically: its precision, the reciprocal variance, exhausts the information in the sample.

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