The best way to combine two unbiased estimates

Question

Two independent labs each report an unbiased estimate of the same physical constant. Lab A's estimate $T_1$ has variance $\sigma_1^2 = 4$ ; the more careful Lab B's estimate $T_2$ has variance $\sigma_2^2 = 1$ . You combine them linearly as $w T_1 + (1-w)T_2$ , which stays unbiased for any weight $w$ .

Find the weight $w$ on Lab A that minimizes the variance of the combined estimate, and report that minimum variance.

Accepted Answer

Write the variance of the blend. By independence, \operatorname{Var}\big(w T1 + (1-w)T2\big) = w^2\sigma1^2 + (1-w)^2\sigma2^2. Minimize over w. Differentiate and set to zero: 2w\sigma1^2 - 2(1-w)\sigma2^2 = 0 \implies w(\sigma1^2 + \sigma2^2) = \sigma2^2 \implies w^\star = \frac{\sigma2^2}{\sigma1^2 + \sigma2^2}. The best weight on each estimate is proportional to the inverse of its variance: precise estimates get more weight. Plugging in the numbers, w^\star = \frac{1}{4 + 1} = \frac{1}{5} = 0.2. So Lab A gets weight 0.2 and the sharper Lab B gets 0.8. The resulting variance. Substituting w…

The best way to combine two unbiased estimates

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