Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 7 / b / i

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- Solution i

Solution

For an observed random sample with all

x_{i} > 0

, which happens almost surely, independence gives the likelihood function

L (θ) = \prod_{i = 1}^{n} \frac{2 x _{i}}{θ} e^{- x_{i}^{2} / θ} = 2^{n} (\prod_{i = 1}^{n} x_{i}) θ^{- n} exp (- \frac{S}{θ}), S = \sum_{i = 1}^{n} x_{i}^{2} > 0.

Ignoring a term independent of

θ

, the log-likelihood and its derivative are

ℓ (θ) = - n lo g θ - \frac{S}{θ} + C, ℓ^{'} (θ) = \frac{S - n θ}{θ ^{2}} .

This derivative is positive for

0 < θ < S / n

and negative for

θ > S / n

. Thus there is a unique global maximum, and the maximum-likelihood estimator is

θ (X) = \frac{1}{n} i = 1 \sum n X_{i}^{2} .

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