91Ó°ÊÓ

. Is the maximum likelihood estimator for \(\sigma^{2}\) in a normal pdf, where both \(\mu\) and \(\sigma^{2}\) are unknown, asymptotically unbiased?

Suppose that \(W_{1}\) is a random variable with mean \(\mu\) and variance \(\sigma_{1}^{2}\) and \(W_{2}\) is a random variable with mean \(\mu\) and variance \(\sigma_{2}^{2}\). From Example 5.4.3, we know that \(c W_{1}+(1-c) W_{2}\) is an unbiased estimator of \(\mu\) for any constant \(c>0\). If \(W_{1}\) and \(W_{2}\) are independent, for what value of \(c\) is the estimator \(c W_{1}+(1-c) W_{2}\) most efficient?

If \(\hat{\theta}\) is sufficient for \(\theta\), show that any one-to-one function of \(\theta\) is also sufficient for \(\theta\).