Question: Let $T_1$ and $T_2$ be independent unbiased estimators of a parameter $\Theta$.
Assume that $\operatorname{Var}(T_2) = \operatorname{Var}(T_1)$.
Using the MSE critertion, define which is a better estimator for $\Theta^2$:
$T_1^2\text{ or }T_1 * T_2$
So I found out that: $\begin{align} & E(T_{1}^2) = \operatorname{Cov}(T_1,T_1)+E(T_1)*E(T_1) = Var(T_1)+\Theta^2\\ \implies & \operatorname{Bias}(T_1^2) = E(T_1^2)-\Theta^2=Var(T_1) \end{align}$ $\begin{align} & E(T_1 * T_2) = \operatorname{Cov}(T_1,T_2) +E(T_1)*E(T_2) = \Theta^2 \\ \implies & \operatorname{Bias}(T_1*T_2) = 0 \end{align}$ $\begin{align} \operatorname{MSE}(T_{1}^2) &= \operatorname{Var}(T_{1}^2) + \operatorname{Bias}^2(T_1^2) = \operatorname{Var}(T_{1}^2)+ \operatorname{Var}^{2}(T_{1}) \\ \operatorname{MSE}(T_{1}*T_2) &= \operatorname{Var}(T_{1}*T_{2}) + \operatorname{Bias}(T_1*T_2) = ... = 2\Theta^2Var(T_1) +\operatorname{Var}^2(T_1) \end{align} $
But still I don't know how to compare between $\operatorname{Var}(T_{1}^2)$ and $2\Theta^2\operatorname{Var}(T_1)$.
I'm not a LaTeX expert, so I hope that it is somewhat readable...Thanks alot!