Variance of estimators

8 The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) have the distributions \(\mathrm { B } \left( n _ { 1 } , \theta \right)\) and \(\mathrm { B } \left( n _ { 2 } , \theta \right)\) respectively. Two possible estimators for \(\theta\) are $$T _ { 1 } = \frac { 1 } { 2 } \left( \frac { X _ { 1 } } { n _ { 1 } } + \frac { X _ { 2 } } { n _ { 2 } } \right) \text { and } T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { n _ { 1 } + n _ { 2 } } .$$

1. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are both unbiased estimators, and calculate their variances.
2. Find \(\frac { \operatorname { Var } \left( T _ { 1 } \right) } { \operatorname { Var } \left( T _ { 2 } \right) }\). Given that \(n _ { 1 } \neq n _ { 2 }\), use the inequality \(\left( n _ { 1 } - n _ { 2 } \right) ^ { 2 } > 0\) to find which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.

6. When a tree seed is planted the probability of it germinating is \(p\). A random sample of size \(n\) is taken and the number of tree seeds, \(X\), which germinate is recorded.

1. 1. Show that \(\hat { p } _ { 1 } = \frac { X } { n }\) is an unbiased estimator of \(p\).
   2. Find the variance of \(\hat { p } _ { 1 }\). A second sample of size \(m\) is taken and the number of tree seeds, \(Y\), which germinate is recorded. Given that \(\hat { p } _ { 2 } = \frac { Y } { m }\) and that \(\hat { p } _ { 3 } = a \left( 3 \hat { p } _ { 1 } + 2 \hat { p } _ { 2 } \right)\) is an unbiased estimator of \(p\),
2. show that
   1. \(\quad a = \frac { 1 } { 5 }\),
   2. \(\operatorname { Var } \left( \hat { p } _ { 3 } \right) = \frac { p ( 1 - p ) } { 25 } \left( \frac { 9 } { n } + \frac { 4 } { m } \right)\).
3. Find the range of values of \(\frac { n } { m }\) for which $$\operatorname { Var } \left( \hat { p } _ { 3 } \right) < \operatorname { Var } \left( \hat { p } _ { 1 } \right) \text { and } \operatorname { Var } \left( \hat { p } _ { 3 } \right) < \operatorname { Var } \left( \hat { p } _ { 2 } \right)$$
4. Given that \(n = 20\) and \(m = 60\), explain which of \(\hat { p } _ { 1 } , \hat { p } _ { 2 }\) or \(\hat { p } _ { 3 }\) is the best estimator.