1 The random variable \(X\) has the continuous uniform distribution with probability density function $$\mathrm { f } ( x ) = \frac { 1 } { \theta } , \quad 0 \leqslant x \leqslant \theta$$ where \(\theta ( \theta > 0 )\) is an unknown parameter.
A random sample of \(n\) observations from \(X\) is denoted by \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\), with sample mean \(\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).

1. Show that \(2 \bar { X }\) is an unbiased estimator of \(\theta\).
2. Evaluate \(2 \bar { X }\) for a case where, with \(n = 5\), the observed values of the random sample are \(0.4,0.2\), 1.0, 0.1, 0.6. Hence comment on a disadvantage of \(2 \bar { X }\) as an estimator of \(\theta\). For a general random sample of size \(n\), let \(Y\) represent the sample maximum, \(Y = \max \left( X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right)\). You are given that the probability density function of \(Y\) is $$g ( y ) = \frac { n y ^ { n - 1 } } { \theta ^ { n } } , \quad 0 \leqslant y \leqslant \theta$$
3. An estimator \(k Y\) is to be used to estimate \(\theta\), where \(k\) is a constant to be chosen. Show that the mean square error of \(k Y\) is $$k ^ { 2 } \mathrm { E } \left( Y ^ { 2 } \right) - 2 k \theta \mathrm { E } ( Y ) + \theta ^ { 2 }$$ and hence find the value of \(k\) for which the mean square error is minimised.
4. Comment on whether \(k Y\) with the value of \(k\) found in part (iii) suffers from the disadvantage identified in part (ii).