1 The random variable $X$ has probability density function

$$\mathrm { f } ( x ) = \frac { x \mathrm { e } ^ { - x / \lambda } } { \lambda ^ { 2 } } \quad ( x > 0 )$$

where $\lambda$ is a parameter $( \lambda > 0 ) . X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are $n$ independent observations on $X$, and $\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i }$ is their mean.\\
(i) Obtain $\mathrm { E } ( X )$ and deduce that $\hat { \lambda } = \frac { 1 } { 2 } \bar { X }$ is an unbiased estimator of $\lambda$.\\
(ii) $\operatorname { Obtain } \operatorname { Var } ( \hat { \lambda } )$.\\
(iii) Explain why the results in parts (i) and (ii) indicate that $\hat { \lambda }$ is a good estimator of $\lambda$ in large samples.\\
(iv) Suppose that $n = 3$ and consider the alternative estimator

$$\tilde { \lambda } = \frac { 1 } { 8 } X _ { 1 } + \frac { 1 } { 4 } X _ { 2 } + \frac { 1 } { 8 } X _ { 3 } .$$

Show that $\tilde { \lambda }$ is an unbiased estimator of $\lambda$. Find the relative efficiency of $\tilde { \lambda }$ compared with $\hat { \lambda }$. Which estimator do you prefer in this case?

\hfill \mbox{\textit{OCR MEI S4 2010 Q1 [24]}}