= Minimum-variance combination of unbiased sample means
{title2=$w_*=(V_Y-C)/(V_X+V_Y-2C)$}
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> $w^2V_X+(1-w)^2V_Y+2w(1-w)C$. Completing the square gives $w_*=(V_Y-C)/(V_X+V_Y-2C)$ and minimum <variance> $(V_XV_Y-C^2)/(V_X+V_Y-2C)$ when the denominator is positive. These are optimal among fixed linear unbiased combinations. Substituting weights estimated from the same observations need not preserve unbiasedness.
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