Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 1 / c

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- Solution c

Solution

r_{s} = 1/ α_{s}

, the log likelihood is

ℓ (r_{0}) = constant - 3 N lo g r_{0} - \frac{1}{r _{0}} \sum_{s} r_{s} .

Hence

r_{0} = \frac{1}{3 N} s = 1 \sum N \frac{1}{α _{s}} .

The score variance and negative expected Hessian give Fisher information

I_{N} (r_{0}) = \frac{3 N}{r _{0}^{2}} .

Since

r_{s}

is Gamma with mean

3 r_{0}

3 r_{0}^{2}

,

E r_{0} = r_{0}, Var (r_{0}) = \frac{r _{0}^{2}}{3 N} .

The estimator is unbiased and attains the Cramér-Rao lower bound.

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