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 $Y$.