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Suppose I know that $ A = 0.3 \cdot B + 0.7 \cdot C $
And I have these estimates: $$ \hat A = A + \varepsilon $$ $$ \hat B = B + \zeta $$ $$ \hat C = C + \eta $$ For the sake of the argument, $ \varepsilon \space \zeta$ and $\eta$ are non-correlated white noise. I could have some estimate on their variance.
What's a way to get an estimate of $A$?

BlackNinja
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    It depends on what $\varepsilon,$ $\zeta,$ and $\eta$ mean, how they have been measured or estimated, and what assumptions you make about them. Please provide those details. – whuber Aug 09 '23 at 20:15
  • Thanks whuber, edited that – BlackNinja Aug 09 '23 at 20:21
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    We have two independent and unbiased estimators for $A$, namely $0.3\hat B + 0.7\hat C$ and $\hat{A}$. If you knew the variance of your noise parameters, you could use precision weighting to combine the estimates: https://en.wikipedia.org/wiki/Inverse-variance_weighting – John Madden Aug 09 '23 at 20:59
  • Got it @JohnMadden. Thanks! – BlackNinja Aug 10 '23 at 05:45

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There are two obvious independent (unbiased) estimates of $A:$ $\hat A$ and $0.3\hat B + 0.7\hat C.$ Their variances are easily expressed in terms of the variances of the three noise terms. The minimum-variance unbiased estimate of $A$ obtained from those two estimates is their weighted average with weights inversely proportional to the variances: see Determining true mean from noisy observations.

whuber
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