91Ó°ÊÓ

For a certain sample the sample variance $$ s^{2}={ }^{n} \sum_{i=1}\left[\left(x_{1},-\bar{x}\right)^{2} / \mathrm{n}\right] $$ is calculated to be 16. The sample contains 9 elements. Using an unbiased estimator. estimate \(\sigma^{2}\)

Show that the sample mean is the minimum variance unbiased linear combination of the observations in a random sample.

Let \(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\) be a random sample from the one-parameter Cauchy distribution. Show that the sample mean, \(\bar{x}\), is not a mean-square error consistent estimator of the parameter \(\theta\).