Question

T is an unbiased estimator for theta, show that
T^2 is a biased estimator for theta^2 .

Answer 1

We have by assumption, E(T)=$\theta$..........(Eq. 1)

Hence, $E(T^2)=Var(T)+(E(T))^2$

Substuting Eq. 1 in above equation we get,

$E(T^2)=Var(T)+\theta^2$.................(Eq. 2)

$\therefore\ Var(T)=E(T^2)-\theta^2$ ...........(Eq. 3)

But since Var(T)>0

$\therefore\ E(T^2)-\theta^2>0$

$\therefore E(T^2)>\theta^2$

Which means, $E(T^2) \not=\theta^2$

Hence we can say that $T^2$ is a biased estimator of $\theta^2.$