Chapter 8.2, Problem 7E

Consider the situation described in Exercise 8.6. How should the constant a be chosen to minimize the variance of ${\hat{\theta }}_3$ if ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are not independent but are such that Cov $\left({\hat{\theta }}_1,{\hat{\theta }}_2\right)=c\ne 0$?

8.6 Suppose that $E\left({\hat{\theta }}_1\right)=E\left({\hat{\theta }}_2\right)=\theta$, $V\left({\hat{\theta }}_1\right)={\sigma }_1^2$, and $V\left({\hat{\theta }}_2\right)={\sigma }_2^2$. Consider the estimator ${\hat{\theta }}_3=a{\hat{\theta }}_1+\left(1-a\right){\hat{\theta }}_2$.

1. a. Show that ${\hat{\theta }}_3$ is an unbiased estimator for θ.
2. b. If ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ are independent, how should the constant a be chosen in order to minimize the variance of ${\hat{\theta }}_3$?