Suppose that it is known that the mean of RV $X_i$ is $\mu_i\theta$, (i = 1, 2,..., n), where $\mu_i$ are known constants, whereas $\theta$ is unknown. Let $\Sigma$ be the variance matrix of the random vector $X = \left[X_1, X_2,\ldots, X_n\right]$.
Show that the minimum-variance unbiased linear estimator of $\theta$ is given by $\hat \theta\left(x\right) = \frac{\mu^T\Sigma^{-1}x}{\mu^T\Sigma^{-1}\mu}$ where $\mu = \left[\mu_1, \mu_2, \ldots, \mu_n \right]$.
For the unbiased part, it is easily to proved. But for the minimum-variance part, I don't have the PDF of $X_i$ so it cannot be proved using traditional way. Anyone has suggestions on this problem?
[self-study]
tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$