Statistics Parameter Estimators

Question

I've been struggling to figure out how to do the following question:

Can anyone help me figure this out?

Thanks.

Do you understand the questions? If yes, what is it exactly about this problem that gives you trouble? — Stijn, Dec 03 '13 at 10:26
Yes, but I'm not quite sure how to calculate E() of theta to check for bias... — Richard, Dec 03 '13 at 18:03

score 0 · Answer 1 · answered Dec 06 '13 at 03:02

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,

$E(\hat{\theta}) - \theta = \sigma^{2}/n$.

1 Answers1