Optimally pooling two estimators

MediumRandom Variables Statistics~5m

Two independent desks each produce an unbiased estimate of the same fair value: estimator $A$ with variance $4$ , estimator $B$ with variance $12$ . You pool them into $\hat\theta=wA+(1-w)B$ . Choosing $w$ to minimize the variance of $\hat\theta$ , what is the smallest achievable variance? Give a decimal to two places.

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Minimize $4w^2+12(1-w)^2$ over $w$ ; optimum weights by inverse variance.
Min variance $=\big(1/\operatorname{Var}A+1/\operatorname{Var}B\big)^{-1}$ .

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3 (± 0.001)

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Optimally pooling two estimators

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