Solving a variance with two estimators and known covariance

Question

I want to solve the following variance of an estimator but after many trials I still didn't succeed...

$V(a\hat{\theta}_1 + (1-a)\hat{\theta}_2)$

We have that:

• $V(\hat θ_1) = σ_1^2$
• $V(\hat θ_2) = σ_2^2$
• $\operatorname{Cov}(\hat θ_1, \hat θ_2) = c ≠ 0$

Any idea?

Answer 1

$\newcommand{\v}{\operatorname{var}} \newcommand{\c}{\operatorname{cov}}$ \begin{align} & \v(a\hat{\theta}_1 + (1-a)\hat{\theta}_2) \\[6pt] = {} & \v(a\hat\theta_1) + \v((1-a)\hat\theta_2) + 2\c(a\hat\theta_1,(1-a)\hat\theta_2) \\[6pt] = {} & a^2 \v(\hat\theta_1) + (1-a)^2\v(\hat\theta_2) + 2a(1-a)\c(\hat\theta_1,\hat\theta_2) \\[4pt] = {} & a^2\sigma_1^2 + (1-a)^2\sigma_2^2 + 2a(1-a)c. \end{align} (A standard exercise asks you to find the value of $a$ that minimizes this.)