Small area estimation methods under cut-off sampling
Section 2. Cut-off sampling in small domains

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We consider a population $U$ partitioned into $m$ subsets $U_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ called hereafter domains or areas, of sizes $N_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ with $N\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m}{N}_i.$ We restrict ourselves to the case in which the domains act as sampling strata. Then, independent samples are drawn from the different domains, where the sample $s_i$ of size $n_i$ from domain $i$ is supposed to be drawn by cut-off sampling, $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$ This is done by excluding a subset of units $U_{iE}\subseteq U_i$ from the selection. In other words, the domain $U_i$ is partitioned into two subsets, $U_{iI}$ and $U_{iE},$ of known sizes $N_{iI}$ and $N_{iE}$ respectively, with $N_i\mathrm{<mo>=</mo>}{N}_{iI}+N_{iE}.$ The set $U_{iI}$ contains the units that can be potentially selected for the sample, called here the set of included units, whereas $U_{iE}$ contains the excluded units.

Let $y_{ij}$ be the value of the target variable $y$ for the $j^{\text{th}}$ unit within the $i^{\text{th}}$ domain. We focus on estimation of domain totals $Y_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{N_i}}{y}_{ij}$ or means ${\overline{Y}}_i\mathrm{<mo>=</mo>}{Y}_i/N_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$ Under cut-off sampling within each domain, the sample $s_i$ is supposed to be drawn from the subset of included individuals, $U_{iI},$ from domain $i.$ Then, the inclusion probabilities for the included individuals $\left(j\in U_{iI}\right)$ are ${\pi }_{j|i}\mathrm{<mo>=</mo>}\text{Pr}\left(j\in s_i\right)\mathrm{<mo>></mo>}0$ and $w_{j|i}\mathrm{<mo>=</mo>}{\pi }_{j|i}^{-1}$ are the corresponding sampling weights. For the excluded units $\left(j\in U_{iE}\right),$ the inclusion probabilities are zero and, therefore, the corresponding sampling weights are not defined. As a consequence, for domains $i$ with $U_{iE}\ne \varnothing ,$ basic design-based estimators of $Y_i$ or ${\overline{Y}}_i$ are biased and a design-unbiased estimator does not exist.

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