8. A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)
- Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\)
An estimator for \(\mu\) is
$$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
- Show that \(\bar { X }\) is a consistent estimator for \(\mu\).
An estimator of \(\sigma ^ { 2 }\) is
$$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
- Find the bias of \(U\).
- Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).