 Minimum-variance combination of unbiased sample means

For two unbiased estimators $A,B$ of the same parameter with variances $V_X,V_Y$ and covariance $C$, the fixed-weight estimator $wA+(1-w)B$ is unbiased and has variance $w^2{V}_X+(1-w{)}^2{V}_Y+2w(1-w)C$. Completing the square gives $w_{\ast }=(V_Y-C)/(V_X+V_Y-2C)$ and minimum variance $(V_X{V}_Y-C^2)/(V_X+V_Y-2C)$ when the denominator is positive. These are optimal among fixed linear unbiased combinations. Substituting weights estimated from the same observations need not preserve unbiasedness.