2. Definitions and notation

Piero Demetrio Falorsi and Paolo Righi

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In this section, we introduce the concepts of estimation domain and planned domain which play a key role in the framework presented herein.

Let $U$  be the reference population of $N$  elements and let $U_d\left(d=1,\dots ,D\right)$  be an estimation domain, i.e., a generic sub-population of $U$  with $N_d$  elements, for which separate estimates must be calculated. Let $y_{rk}$  denote the value of the $r^{\text{th}}$   $\left(r=1,\dots ,R\right)$  variable of interest attached to the $k^{\text{th}}$  population unit and let ${\gamma }_{dk}$  denote the domain membership indicator for unit $k$  defined as

$${\gamma }_{dk}=\{\begin{array}{cc}1& \text{if }k\in U_d\\ 0& \text{otherwise}\end{array}.(2.1)$$

We assume that the ${\gamma }_{dk}$ values are available in the sampling frame and more than one value ${\gamma }_{dk}$   $\left(d=1,\dots ,D\right)$  can be 1 for each unit $k;$  therefore, the estimation domains can overlap.

The parameters of interest are the $D\times R$  domain totals

$$t_{\left(dr\right)}={\displaystyle {\sum }_{k\in U}{y}_{rk}{\gamma }_{dk}\left(r=1,\dots ,R;d=1,\dots ,D\right).(2.2)}$$

Let $p(\cdot )$  be a single-stage without replacement sampling design and $\pi ={\left({\pi }_1,\dots ,{\pi }_k,\dots ,{\pi }_N\right)}^{\prime }$  be the $N­$  vector of inclusion probabilities. Let $s$  be the sample selected with probability $p\left(s\right).$  Denote by $U_h$   $\left(h=1,\dots ,H\right)$  the subpopulation of size $N_h={\displaystyle {\sum }_{k\in U_h}{\delta }_{hk}}$  where ${\delta }_{hk}=\text{1 if }k\in U_h$  and ${\delta }_{hk}=0$  otherwise.

We focus on fixed size sampling designs which are those satisfying

$${\displaystyle {\sum }_{k\in s}{\delta }_k}=n,(2.3)$$

where ${\delta }_k={\left({\delta }_{1k},\dots ,{\delta }_{hk},\dots ,{\delta }_{Hk}\right)}^{\prime }$  and $n={\left(n_1,\dots ,n_h,\dots ,n_H\right)}^{\prime }$  is the vector of integer numbers defining the sample sizes fixed at the design stage. Since the sample size $n_h,$  corresponding to $U_h,$  does not vary among sample selections, the subpopulation $U_h$  will be referred to as a planned domain in the sequel. A necessary but not sufficient condition for ensuring that (2.3) is satisfied is that the vector $\pi$  is such that

$${\displaystyle {\sum }_{k\in U}{\pi }_k{\delta }_k}=n.(2.4)$$

In our setting, the planned domains can overlap; therefore, the unit $k$  may have more than one value ${\delta }_{hk}=1$  (for $h=1,\dots ,H).$ Let us suppose that the ${\delta }_{hk}$  values are known, and available in the sampling frame, for all population units. We suppose furthermore that the $N\times H$  matrix ${\left({\delta }_1,\dots ,{\delta }_k,\dots ,{\delta }_N\right)}^{\prime }$  is non-singular.

The planned domains and their relationship with the estimation domains play a central role in our generalized framework. We assume that the estimation domains may be defined as an aggregation of complete planned domains, which ensure that the expected sample size in the $d^{\text{th}}$  estimation domain $U_d,$  say $n_d,$  can be obtained as a simple aggregation of the expected sample sizes of the planned domains that are included within it. Finally, let ${\widehat{t}}_{\left(dr\right)}$  be the Horvitz-Thompson (HT) estimator of $t_{\left(dr\right)}$  with

$${\widehat{t}}_{\left(dr\right)}={\displaystyle {\sum }_{k\in s}\frac{1}{{\pi }_k}{y}_{rk}{\gamma }_{dk}}.(2.5)$$

An example from business surveys. Suppose that the survey estimates must be calculated separately considering three domain types: region (with 20 modalities), economic activity (2 modalities: goods and services) and enterprise size (3 modalities: small, medium and large enterprises). That is, there are $D=20+2+3=25$  possible overlapping estimation domains. The planned domains can be defined with different options.

Option 1. The single planned domain $U_h$  is identified by a specific intersection of the categories of the estimation domains. In this case $H=20\times 2\times 3=120$  planned domains are defined. They represent a specific partition of $U.$  The planned domains do not overlap and ${\displaystyle {\sum }_h{\delta }_{hk}=1}.$

Option 2. The planned domains $U_h$  coincide with the estimation domains. Therefore, $H=D=25$  and the ${{\delta }^{\prime }}_k$  are defined as vectors with three 1’s, so that ${\displaystyle {\sum }_h{\delta }_{hk}=3}.$  Recall that the planned domains overlap.

Option 3. The planned domains $U_h$  are defined as (i) region by economic activity and (ii) economic activity by enterprise size; then, $H=(20\times 2)+(2\times 3)=46$  with ${\displaystyle {\sum }_h{\delta }_{hk}=2}.$

Other intermediate relationships among estimation and planned domains are possible.

It is emphasised that the planned domains represent the basis for defining broad classes of sampling designs. For instance, stratified sampling designs require that the planned domains do not overlap, as ${\displaystyle {\sum }_h{\delta }_{hk}=1}$  and each $U_h$  is referred to as a stratum. Therefore, Option 1 in the example above leads us to define a stratified sampling design. Furthermore, the strata defined as in Option 1 are the basis of the so-called “multi-way stratified sampling design” (Winkler 2001).

If ${\displaystyle {\sum }_h{\delta }_{hk}>1},$  the sample sizes of the planned domains identified in Option 1 (strata) are not strictly controlled. Nevertheless, the sample sizes are still controlled at an aggregated level. In Option 2 of the example above, the sample sizes are controlled only for the estimation domains; while in Option 3, the sample sizes are controlled for the subsets of two different partitions, defined by (i) the region by economic activity and (ii) the economic activity by enterprise size. On the basis of the Winkler’s definition, we denote the designs using these types of planned domains as Incomplete multi-way Stratified Sampling (ISS) designs.

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