6. A random sample \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 2 n }\) is taken from a population with mean \(\frac { \mu } { 3 }\) and variance \(3 \sigma ^ { 2 }\). A second random sample \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , \ldots , Y _ { n }\) is taken from a population with mean \(\frac { \mu } { 2 }\) and variance \(\frac { \sigma ^ { 2 } } { 2 }\), where the \(X\) and \(Y\) variables are all independent.
\(A\), \(B\) and \(C\) are possible estimators of \(\mu\), where
$$\begin{aligned}
& A = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } + Y _ { 1 } + Y _ { 2 } } { 2 }
& B = \frac { 3 X _ { 1 } } { 2 } + \frac { 2 Y _ { 1 } } { 3 }
& C = \frac { 3 X _ { 1 } + 4 Y _ { 1 } } { 3 }
\end{aligned}$$
- Show that two of \(A , B\) and \(C\) are unbiased estimators of \(\mu\) and find the bias of the third estimator of \(\mu\).
- Showing your working clearly, find which of \(A\), \(B\) and \(C\) is the best estimator of \(\mu\).
The estimator
$$D = \frac { 1 } { k } \left( \sum _ { i = 1 } ^ { 2 n } X _ { i } + \sum _ { i = 1 } ^ { n } Y _ { i } \right)$$
is an unbiased estimator of \(\mu\).
- Find \(k\) in terms of \(n\).
- Show that \(D\) is also a consistent estimator of \(\mu\).
- Find the least value of \(n\) for which \(D\) is a better estimator of \(\mu\) than any of \(A\), \(B\) or \(C\).