Chapter 5.7, Problem 4Q

An estimator ${\hat{\theta }}_n$ is said to be squared-error consistent for $\theta$ if $\underset{n\to \infty }{\lim }E\left[{\left({\hat{\theta }}_n-\theta \right)}^2\right]=0$

• (a) Show that any squared-error consistent ${\hat{\theta }}_n$ is asymptotically unbiased (see Question 5.4.15).

• (b) Show that any squared-error consistent ${\hat{\theta }}_n$ is consistent in the sense of Definition 5.7.1.