Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 17H / c

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- Solution c

Solution

Put

S_{r} = \sum_{i = 1}^{n} x_{i}^{r} .

Independence and

var (Y_{i}) = θ x_{i}

give

var (θ_{L S}) = \frac{θ S _{3}}{S _{2}^{2}},

whereas

\sum_{i} Y_{i} \sim Poisson (θ S_{1})

gives

var (θ_{M L E}) = \frac{θ}{S _{1}} .

The Cauchy-Schwarz inequality, applied to

(x_{i}^{1/2})

and

(x_{i}^{3/2})

, yields

S_{2}^{2} = (\sum_{i} x_{i}^{1/2} x_{i}^{3/2})^{2} \leq S_{1} S_{3} .

Consequently

var (θ_{M L E}) \leq var (θ_{L S}) .

Equality holds exactly when all positive exposures

x_{i}

are equal. This is the variance comparison for Poisson exposure estimators.

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