Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$.
a) Show that for any constant $c$, the weighted average theta $\hat\theta_3= c\cdot\hat\theta_1 +(1-c)\cdot\hat\theta_2$ is an unbiased estimator of $\theta$.
b) Find the $c$ for which $\hat\theta_3$ has the smallest MSE.
c) Are there any values for $c$ $(0\le c\le 1)$ for which $\hat\theta_3$ is better (in the sense of MSE) than both $\hat\theta_1$ and $\hat\theta_2$?