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I am reading a paper on dual frame surveys which has confused me.

Without going too deep into the notation, Hartley has an estimator for dual frame surveys which we will denote $\hat{Y}(\theta)$. This takes both samples from the two frames and performs a weighted average estimate for the overlap domain $Y_{ab}$.

Now Fuller and Burmeister (retrieved here) suggests that if we take additional measurements in our survey such as $x_i$ then incorporating the difference $\hat{X}_{ab}^A-\hat{X}_{ab}^B$ into this estimate could reduce the variance, where $\hat{X}_{ab}^A$ is the estimated total $X$ for the overlap of the two frames using the sample from frame $A$, etc.

My question first of all is whether or not this is true? and if so, why?

I can't understand why adding additional terms which do not measure $y_i$ would help reduce the variance of our estimate of the population total $Y$.

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Consider two variables $U$ and $V$. From basic properties of variances,

$\text{Var}(U-V) = \text{Var}(U) + \text{Var}(V) - 2 \text{Cov}(U,V)$

So if $ \text{Cov}(U,V)>Var(V)/2$ then $\text{Var}(U-V) < \text{Var}(U)$

So if $E(V)=0$ but $\text{Cov}(U,V)$ is reasonably large, $U-V$ has the same expectation but smaller variance than $U$.

The trick is to construct some $V$ that is sufficiently positively related (or equivalently a $W=-V$ that is sufficiently negatively related to add) to the quantity we're using as our original estimator.

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