Statistics Parameter Estimators I've been struggling to figure out how to do the following question:
question http://puu.sh/5AIRS.png
Can anyone help me figure this out?
Thanks.

For part (a), the key is to use the fact that
$E(\bar{X}_{n}^{2}) = E(\bar{X}_{n})^{2} + \textrm{var}(X_{n}) = \mu^{2} + \sigma^{2}/n$
It then follows that
$E(\hat{\theta}) = E(\bar{X}_{n}^{2}) - 4E(X_{n}) + 4
=  \mu^{2} + \sigma^{2}/n - 4\mu + 4
= (\mu - 2)^{2} + \sigma^{2}/n$.
Hence,
$E(\hat{\theta}) - \theta = \sigma^{2}/n$.
