6. A random sample of size \(n\) is taken from the random variable \(X\), which has a continuous uniform distribution over the interval [ \(0 , a\) ], \(a > 0\)
The sample mean is denoted by \(\bar { X }\)
- Show that \(Y = 2 \bar { X }\) is an unbiased estimator of \(a\)
The maximum value, \(M\), in the sample has probability density function
$$f ( m ) = \left\{ \begin{array} { c c }
\frac { n m ^ { n - 1 } } { a ^ { n } } & 0 \leqslant m \leqslant a
0 & \text { otherwise }
\end{array} \right.$$ - Find E(M)
- Show that \(\operatorname { Var } ( M ) = \frac { n a ^ { 2 } } { ( n + 2 ) ( n + 1 ) ^ { 2 } }\)
The estimator \(S\) is defined by \(S = \frac { n + 1 } { n } M\)
Given that \(n > 1\) - state which of \(Y\) or \(S\) is the better estimator for \(a\). Give a reason for your answer.