Reducing the response imbalance: Is the accuracy of the survey estimates improved? Section 6. Statistical properties of the CAL estimator deviationReducing the response imbalance: Is the accuracy of the survey estimates improved? Section 6. Statistical properties of the CAL estimator deviation

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In the decomposition (5.2), the deviation of CAL from the unbiased FUL is ${\widehat{Y}}_{CAL}-{\widehat{Y}}_{FUL}=\widehat{N}{\Delta }_r$ where ${\Delta }_r={\left(b_r-b_s\right)}^{\prime }{\overline{x}}_s.$ To see if ${\Delta }_r$ is smaller, or likely to be so, by realizing low imbalance in data collection, we seek analytic results about statistical properties, such as mean and variance, of ${\Delta }_r$ as a function of the $IMB$ statistic (3.2). Highly general results of this kind are hard to obtain. Several factors complicate the analysis, such as the sampling design used to draw $s,$ the probability distribution of the response sets $r$ given $s,$ the make-up of the auxiliary vector $x,$ and so on. Results for special situations are obtained in Sections 7 and 8.

Result 1 in Section 7 gives properties $–$ expected value and variance $–$ of ${\Delta }_r={\left(b_r-b_s\right)}^{\prime }{\overline{x}}_s$ over response outcomes $r$ with fixed size $m$  and fixed mean ${\overline{x}}_r$ when $x$ is a group vector, and $s$ is a simple random sample. The mean of ${\Delta }_r$ over such outcomes is zero. The imbalance appears in the variance of ${\Delta }_r,$ which is linearly increasing in $IMB,$ approximately. A reason for taking $x$ to be a group vector is that conditioning on ${\overline{x}}_r$ grants relatively simple derivations. A fixed ${\overline{x}}_r$ implies a fixed value $IMB.$ (But the opposite is not true; several ${\overline{x}}_r$ can give the same $IMB.)$ Another simplification when $x$ is a group vector is due to diagonal matrices ${\Sigma }_r$ and ${\Sigma }_s.$ The empirical test in Section 9.1 addresses Result 1.

Simple derivations for the group vector are at the expense of generality. The $x‑$vectors used in production at Statistics Sweden, for example, are often not group vectors. To get transparent mathematical results about ${\Delta }_r$ is then more difficult.

Result 2 in Section 8 is derived under a model of linear regression between $y$ and $x.$ The $y_k$ are then considered random, with properties stated by the model. A group vector feature for $x$ is no longer necessary. The conclusions are in some respects similar to those in Result 1. The empirical Test situation 2 in Section 9.2 refers to both Results 1 and 2.

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