Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 7 / a / iii

Table of contents
- Solution iii

Solution

Differentiating the cumulative distribution function in part (ii) gives the probability density function

n t^{n - 1}

on

(0, 1)

. Hence

E (\frac{M}{θ}) = \int_{0}^{1} t n t^{n - 1} d t = \frac{n}{n + 1}, E M = \frac{n}{n + 1} θ .

Thus an unbiased estimator based on the sample maximum is

θ = \frac{n + 1}{n} θ .

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