Is Square of unbiased estimator is unbiased again?

Question

I am kind of new in here.

I have a question that bothering me that the square of an unbiased estimator is an unbiased estimator. I know that it is not the case but how I can prove that? Using $V(o) = E[o^2] - (E[o])^2$ or some other technique.

Answer 1

Your question is that, if $\hat{\theta}=E(\theta)$, where $\theta$ is a parameter in a statistical distribution($\hat{\theta}$ is its unbaised estimator), then why $\hat{\theta}^2\neq E(\theta^2)$? (i.e. Why $\hat{\theta}^2$ is NOT unbaised)

This can be seen by applying the formula $$ Var(\theta^2)=E(\theta^2)-[E(\theta)]^2\geq0. $$ So $$ E(\theta^2)> [E(\theta)]^2 $$ when $Var(\theta)\neq0$. So in general, $\hat{\theta}^2=[E(\theta)]^2\neq E(\theta^2)$.