Standard +0.3 This is a straightforward application of expectation and variance properties to assess estimator bias and efficiency. Students need to apply E(aX+bY) and Var(aX+bY) linearly, then compare—all standard S4 techniques with no novel insight required. Slightly easier than average due to mechanical nature.
\begin{enumerate}
\item A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }$ is taken from a population with mean $\mu$ and variance $\sigma ^ { 2 }$.\\
(a) Determine the bias, if any, of each of the following estimators of $\mu$.
\end{enumerate}
$$\begin{aligned}
& \theta _ { 1 } = \frac { X _ { 3 } + X _ { 4 } + X _ { 5 } } { 3 } \\
& \theta _ { 2 } = \frac { X _ { 10 } - X _ { 1 } } { 3 } \\
& \theta _ { 3 } = \frac { 3 X _ { 1 } + 2 X _ { 2 } + X _ { 10 } } { 6 }
\end{aligned}$$
(b) Find the variance of each of these estimators.\\
(c) State, giving reasons, which of these three estimators for $\mu$ is\\
(i) the best estimator,\\
(ii) the worst estimator.\\
\hfill \mbox{\textit{Edexcel S4 2008 Q1 [13]}}