Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 28J / a

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- Solution a

Solution

The log-likelihood is, up to constants,

n α lo g β - β \sum_{i} X_{i}

, so the maximum-likelihood estimator is

β = α / \overline{X}

. By the central limit theorem,

n (\overline{X} - α / β) \Rightarrow N (0, α / β^{2}) .

Applying the delta method to

g (x) = α / x

gives

n (β - β) \Rightarrow N (0, β^{2} / α) .

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