Reducing the response imbalance: Is the accuracy of the survey estimates improved? Section 8. The second resultReducing the response imbalance: Is the accuracy of the survey estimates improved? Section 8. The second result

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In Result 1, the survey variable values $y_k$ are treated as fixed, nonrandom. In Result 2, they are random with properties as stated in a linear regression model $\xi$ with residuals ${\epsilon }_k=y_k-x_k^{\prime }\beta$ for some unknown  $\beta :$

$$E_{\xi }\left(y_k|x_k\right)=x_k^{\prime }\beta ;E_{\xi }\left({\epsilon }_k^2|x_k\right)={\sigma }_{\epsilon }^2,\text{all}k\in s;E_{\xi }\left({\epsilon }_k{\epsilon }_{\mathcal{l}}|x_k,x_{\mathcal{l}}\right)=0,\text{all}k\ne \mathcal{l}\in s.(8.1)$$

The properties in (8.1) apply also to units $k$ and $\mathcal{l}$ belonging in any subset $r$ of $s.$ Result 2 presents expected value and approximate variance of ${\Delta }_r={\left(b_r-b_s\right)}^{\prime }{\overline{x}}_s$ conditionally on a fixed self-weighting sample $s$ and a fixed response set $r$ with respective sizes $n$ and $m.$

Result 2: Let $s$ of size $n$ be a self-weighting sample. Let $X$ be the $J\times n$ $x‑$data matrix with columns $x_k,$ $k\in s.$ Then, under the model $\xi$ in (8.1),

$$E_{\xi }\left({\Delta }_r|X,r,s\right)=0;E_{\xi }\left({\Delta }_r^2|X,r,s\right)\approx \left(1-p+\frac{IMB}{p^2}\right)\frac{{\sigma }_{\epsilon }^2}{m},(8.2)$$

where $m$ is the size of the fixed response set $r,$ $p=m/n$ is the response rate and $IMB$ is given by (3.2).

Result 2 (for arbitrary $x‑$vector and random $y_k)$ mirrors Result 1 (for group $x‑$vector and non-random $y_k)$ in that both give conditional mean zero and the same linearly increasing form for the conditional variance approximation.

The derivation in Appendix 3 of Result 2 relies on a comparison of the two quadratic forms in ${\overline{x}}_r-{\overline{x}}_s$ given in (3.1), $Q_s$ and $Q_r.$ The former is used in the imbalance statistic (3.2), $IMB=P^2{Q}_s;$ the latter determines the weight factors $g_k$ for the CAL estimator (5.1). The approximation $Q_r\approx Q_s,$ needed for Result 2, is justified in Appendix 2.

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