- Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\)
\(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\)
Korhan's estimator is given by
$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$
Louise's estimator is given by
$$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$
- Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
- Find \(\operatorname { Var } ( K )\)
- Find \(\operatorname { Var } ( L )\)
The winner of the challenge is the person who finds the better estimator.
- Determine the winner of the challenge for large values of \(n\).
Give reasons for your answer.