1 A parcel is weighed, independently, on two scales. The weights are given by the random variables \(W _ { 1 }\) and \(W _ { 2 }\) which have underlying Normal distributions as follows.
$$W _ { 1 } \sim \mathrm {~N} \left( \mu , \sigma _ { 1 } ^ { 2 } \right) , \quad W _ { 2 } \sim \mathrm {~N} \left( \mu , \sigma _ { 2 } ^ { 2 } \right) ,$$
where \(\mu\) is an unknown parameter and \(\sigma _ { 1 } ^ { 2 }\) and \(\sigma _ { 2 } ^ { 2 }\) are taken as known.
- Show that the maximum likelihood estimator of \(\mu\) is
$$\hat { \mu } = \frac { \sigma _ { 2 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 1 } + \frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } W _ { 2 } .$$
[You may quote the probability density function of the general Normal distribution from page 9 in the MEI Examination Formulae and Tables Booklet (MF2).]
- Show that \(\hat { \mu }\) is an unbiased estimator of \(\mu\).
- Obtain the variance of \(\hat { \mu }\).
- A simpler estimator \(T = \frac { 1 } { 2 } \left( W _ { 1 } + W _ { 2 } \right)\) is proposed. Write down the variance of \(T\) and hence show that the relative efficiency of \(T\) with respect to \(\hat { \mu }\) is
$$y = \left( \frac { 2 \sigma _ { 1 } \sigma _ { 2 } } { \sigma _ { 1 } ^ { 2 } + \sigma _ { 2 } ^ { 2 } } \right) ^ { 2 }$$
- Show that \(y \leqslant 1\) for all values of \(\sigma _ { 1 } ^ { 2 }\) and \(\sigma _ { 2 } ^ { 2 }\). Explain why this means that \(\hat { \mu }\) is preferable to \(T\) as an estimator of \(\mu\).