In several situations, I have two unbiased estimators, and I know one of them is better (lower variance) than the other. However, I would like to get as much information as possible, and I would like to do better than throwing out the weaker estimator.
$$Outcome = Skill + Luck$$$$\newcommand{\Outcome}{\text{Outcome}}\newcommand{\Skill}{\text{Skill}}\newcommand{\Luck}{\text{Luck}}\Outcome = \Skill + \Luck$$
$Outcome$$\Outcome$ is observed. $Skill$$\Skill$ is what I would like to determine. $Luck$$\Luck$ is known to have the average value $0$. From other observables, I can estimate $Luck$$\Luck$ by $L$ so that $E(L) = 0$$\mathbb E(L) = 0$ and $\text{Var}(Luck-L) \lt \text{Var}(Luck)$$\mathrm{Var}(\Luck-L) \lt \mathrm{Var}(\Luck)$.
$Outcome$$\Outcome$ is an unbiased estimator for $Skill$$\Skill$. A better estimate from variance reduction is $Outcome - L$$\Outcome - L$, which is also unbiased. For example, in one situation $Outcome$$\Outcome$ is the average of repeated trials, and I might produce a $95\%$ confidence interval of $[-5.0,13.0]$ without using variance reduction. Using variance reduction, I might get a confidence interval of $[-2.0,4.0]$.
The typical practice is for people to use $Outcome-L$$\Outcome-L$ instead of $Outcome$$\Outcome$. However, this is unsatisfactory to me because in my experience, there is more information in the pair $(Outcome, Outcome-L)$$(\Outcome, \Outcome-L)$ than in just $Outcome-L$$\Outcome-L$. Specifically, in some situations I know that if $Outcome$$\Outcome$ is low, then $Outcome-L$$\Outcome-L$ tends to be an underestimate for $Skill$$\Skill$, and if $Outcome$$\Outcome$ is high, then $Outcome-L$$\Outcome-L$ tends to be an overestimate for $Skill$$\Skill$.
#What's a good way to take advantage of the extra information from knowing both estimators?What's a good way to take advantage of the extra information from knowing both estimators?
