5. (a) Explain briefly what you understand by
- an unbiased estimator,
- a consistent estimator.
of an unknown population parameter \(\theta\).
From a binomial population, in which the proportion of successes is \(p , 3\) samples of size \(n\) are taken. The number of successes \(X _ { 1 } , X _ { 2 }\), and \(X _ { 3 }\) are recorded and used to estimate \(p\).
(b) Determine the bias, if any, of each of the following estimators of \(p\).
$$\begin{aligned}
& \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n }
& \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n }
& \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n }
\end{aligned}$$
(c) Find the variance of each of these estimators.
(d) State, giving a reason, which of the three estimators for \(p\) is - the best estimator,
- the worst estimator.