I have two scalars $\hat{\beta}_1$ and $\hat{\beta}_2$ which are unbiased estimators of $\beta$. Furthermore, the covariance matrix is given by $$var\begin{pmatrix}\hat{\beta_1}\\ \hat{\beta_2} \end{pmatrix}=\begin{pmatrix}\sigma^2_1&0\\0&\sigma^2_2\end{pmatrix}.$$ The ratio $\sigma_2/\sigma_1$ is known. Now, I have to construct a more efficient unbiased estimator of $\beta$. Could someone help me out here? I am fairly sure I should use Generalised least squares, but I don't know in what form. Thanks in advance!
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Hi: divide the elements in the matrix by $\sigma^2_1$. Then, the covariance matrix becomes $\sigma^2_1 \times \left[ {\begin{array} {cc} 1 & 0 \\ 0 & \frac{\sigma^2_{2}}{\sigma^2_{1}} \\ \end{array} } \right] $
so that $\sigma^2_{1}$ is just a scale factor. Now, you can solve the problem:
minimize $w_1^2 \times 1 + w_2^2 \times \frac{\sigma^2_2}{\sigma^2_1}$
subject to $w_1 + w_2 = 1 $.
The resulting estimator, $w_1 \times \hat\beta_1 + w_2 \times \hat\beta_2$ is unbiased and minimum variance. You can use the lagrange multiplier approach to deal with the constraint or substitute $1-w_1$ for $w_2$ and take the derivative.
ADDENDUM A: Use the second approach.
This results in $w_1^2 + (1-w_1)^2 \times \frac{\sigma^2_2}{\sigma^2_1}$
Taking the derivative gives: $2 \times w_1 - 2 \times (1 - w_1) \times \frac{\sigma^2_2}{\sigma^2_1} = 0.$. This leads to
$2 \times w_1 + 2 \times w_1 \times \frac{\sigma^2_2}{\sigma^2_1} = 2 \times\frac{\sigma^2_2}{\sigma^2_1} $
$w_1 \times \left (2 + 2 \times \frac{\sigma^2_2}{\sigma^2_1} \right) = 2 \times \frac{\sigma^2_2}{\sigma^2_1}$
Please check the algebra because I expected something simpler for $w_1$.
ADDENDUM B: Actually, for $\sigma^2_1 = \sigma^2_2$, $w_1 = \frac{1}{2}$ which is correct (give same weight to both estimators) so the algebra is probably fine.
