Let $T_1$ be an unbiased estimator of $\theta$. Show that $T_1^2$ is a biased estimator of $\theta^2$.

Question

Let $T$ be an unbiased estimator of $\theta$. Show that $T^2$ is a biased estimator of $\theta^2$.

We have $\operatorname{E}(T)=\theta$ by assumption. Then $\operatorname{E}(T^2)=\operatorname{Var}(T)+(\operatorname{E}(T))^2=\operatorname{Var}(T)+\theta^2$. But we can't guarantee $\operatorname{Var}(T)$ is not zero. How can the question be answered?

Answer 1

If $\operatorname{Var}(T) = 0$, then $T$ is a constant, which means your estimator is a constant random variable that's just "the right answer" ($\theta$) with probability 1. My guess is that it's safe to assume this isn't the case.

(But if it somehow is the case, then yes -- $T^2$ would indeed be an unbiased estimator of $\theta^2$.)