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 $\hat\theta_2$?


For part (a)

$\begin{align} E(\hat{\theta}_{3})&=E(c\hat{\theta}_{1}+(1-c)\hat{\theta}_{2})\\ &=cE(\hat{\theta}_{1})+(1-c)E(\hat{\theta}_{2}) \\ &=c\theta+(1-c) \theta \\ &=c\theta+\theta-c\theta\\ &=\theta \end{align} $

For part(b)

$\begin{align} Var(\hat{\theta}_{3})&= Var(c \hat{\theta}_{1}+(1-c) \hat{\theta}_{2}) \\ &= c^{2} Var(\hat{\theta}_{1}) +(1-c)^2 Var(\hat{\theta}_{2}) \\ &= 2c^2 Var(\hat{\theta}_{2}) +(1-c)^2 Var(\hat{\theta}_{2})\\ &= Var(\hat{\theta}_{2})(2c^2+(1-c)^2)\\ \end{align} $

To find the minimum variance, we take the derivative with respect to $c$ and equate it with zero

$\begin{align} \frac{d}{dc} Var(\hat{\theta}_{3})&= 4c+2(1-c)(-1) =0 \\ &= 4c+-2+2c=0 \end{align} $

Therefore, $c=\frac{1}{3}$


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