# 4 Proposed methods

Pierre Lavallée and Sébastien Labelle-Blanchet

The methods proposed in this section for reducing the variance of the estimates are mainly based on the use of weighted links for the computation of the estimates of $Y$ under Indirect Sampling. We will therefore use estimator (2.9), rather than estimator (2.3). A first set of methods is based on the use of weighted links ${\theta }_{j,i}$ that are proportional to some measure of size for the establishments. The second set of methods uses the optimal solutions presented in Section 2.2, under different assumptions. Finally, the last set of methods considers the use of the exact selection probabilities, rather than the estimation weights obtained from the GWSM, under two sampling scenarios.

## 4.1  Methods based on the use of weighted links

Method 1: ${\theta }_{j,i}$ proportional to ${\pi }_{j}^{A}$

We first propose to reduce the variance (2.11) by setting ${\theta }_{j,i}$ proportional to ${\pi }_{j}^{A}.$ Formally, this can be written as ${\theta }_{j,i}^{\pi }={\pi }_{j}^{A}{l}_{j,i}$. In business surveys, because stratification is usually done by size (according to some size measure), setting ${\theta }_{j,i}$ proportional to ${\pi }_{j}^{A}$ can be viewed as assigning large weights to links of large establishments, and small weights to small ones, which is a natural approach.

With this method, we have ${\stackrel{˜}{\theta }}_{j,i}^{\pi }={\theta }_{j,i}^{\pi }/{\theta }_{i}^{\pi B}={\pi }_{j}^{A}{l}_{j,i}/{\sum }_{j=1}^{{M}^{A}}{\pi }_{j}^{A}{l}_{j,i}.$ Because of the many-to-one correspondence between ${U}^{A}$ and ${U}^{B},$ we obtain ${\stackrel{˜}{\theta }}_{j,i}^{\pi }={\theta }_{j,i}^{\pi }/{\theta }_{i}^{\pi B}=$ ${\pi }_{j}^{A}{l}_{j,i}/{\sum }_{j=1}^{{M}_{i}^{B}}{\pi }_{j}^{A}.$ Therefore, from (2.8), we have

${w}_{j}^{\pi }=\frac{{\sum }_{j=1}^{{M}_{i}^{B}}{t}_{j}^{A}}{{\sum }_{j=1}^{{M}_{i}^{B}}{\pi }_{j}^{A}}.$(4.1)

Using (4.1), we can rewrite estimator (2.9) as

${\stackrel{^}{Y}}_{\pi }=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{Z}_{hj}^{\pi }$(4.2)

where

${Z}_{hj}^{\pi }=\frac{{\pi }_{j}^{A}{Y}_{i}}{{\sum }_{j=1}^{{M}_{i}^{B}}{\pi }_{j}^{A}}$(4.3)

for $j\in h$ and $j\in i.$ It should be noted that if all establishments $j$ of a given enterprise belong to the same stratum $h,$ say, we have ${\stackrel{˜}{\theta }}_{j,i}=1/{M}_{i}^{B},$ and estimator (4.2) is then equivalent to estimator (2.1) (and (2.3)).

For computing the variance of ${\stackrel{^}{Y}}_{\pi },$ we use formula (2.11) with the values (4.3). For the example of Section 3, we get ${V\left(\stackrel{^}{Y}}_{\pi }\right)=$ 439,111, which is a strong reduction compared to $V\left(\stackrel{^}{Y}\right)=$ 1,115,111, but still relatively far from $V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480.

Method 2: ${\theta }_{j,i}$ proportional to some size measure ${x}_{j}$

We propose to reduce the variance (2.11) by setting ${\theta }_{j,i}$ proportional to some size measure $x$ correlated with the variable of interest $y.$ We assume that variable ${x}_{j}$ is available for all establishments $j\in {U}^{A}.$ This variable could be used, for instance, to stratify the sampling frame ${U}^{A}$ by size. As for Method 1, setting ${\theta }_{j,i}^{x}={x}_{j}{l}_{j,i}$ can be viewed as assigning large weights to links of large establishments, and small weights to small ones, which again is a natural approach. With this method, we have ${\stackrel{˜}{\theta }}_{j,i}^{x}={\theta }_{j,i}^{x}/{\theta }_{i}^{xB}={x}_{j}{l}_{j,i}/{\sum }_{j=1}^{{M}^{A}}{x}_{j}{l}_{j,i}.$

We have ${\stackrel{˜}{\theta }}_{j,i}^{x}={\theta }_{j,i}^{x}/{\theta }_{i}^{xB}={x}_{j}{l}_{j,i}/{\sum }_{j=1}^{{M}_{i}^{B}}{x}_{j}={x}_{j}{l}_{j,i}/{X}_{i}$ because of the many-to-one correspondence between ${U}^{A}$ and ${U}^{B}.$ Therefore, from (2.8), we have

${w}_{i}^{x}=\frac{1}{{X}_{i}}\sum _{j=1}^{{M}_{i}^{B}}\frac{{t}_{j}^{A}{x}_{j}}{{\pi }_{j}^{A}}.$(4.4)

Using (4.4), we can rewrite (2.9) as

${\stackrel{^}{Y}}_{x}=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{Z}_{hj}^{x}$(4.5)

where

${Z}_{hj}^{x}=\frac{{Y}_{i}}{{X}_{i}}{x}_{j}$(4.6)

for $j\in h$ and $j\in i.$

To compute the variance of ${\stackrel{^}{Y}}_{x},$ we use formula (2.11) together with (4.6). For the example of Section 3, the variable $x$ corresponds to the number of employees (see Figure 3.1). The correlation between the revenue $y$ and the number of employees $x$ is relatively high $\left(\rho =$ 92.8%). We obtain ${V\left(\stackrel{^}{Y}}_{x}\right)=$ 686,540, which is again a strong reduction compared to $V\left(\stackrel{^}{Y}\right)=$ 1,115,111, but still relatively far from $V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480.

Method 3: ${\theta }_{j,i}$ proportional to the variable of interest ${y}_{j}$

The third method proposed is to reduce the variance (2.11) by setting ${\theta }_{j,i}$ proportional to the variable of interest $y$ measured for the establishment $j$ belonging to enterprise $i.$ Obviously, setting ${\theta }_{j,i}^{y}={y}_{j}{l}_{j,i}$ assigns large weights to links of large establishments, and small weights to small ones, which again is a natural approach. Because ${y}_{j}$ is unknown at the beginning of the survey, this method might not look as being implementable since ${\theta }_{j,i}^{y}$ depends on ${y}_{j}.$ Now, because of the many-to-one correspondence between ${U}^{A}$ and ${U}^{B},$ every quantity entering in ${\theta }_{j,}^{y}{}_{i}$ are measured through the Indirect Sampling process.

The proposed method is feasible in this setting and we have ${\stackrel{˜}{\theta }}_{j,i}^{y}={\theta }_{j,i}^{y}/{\theta }_{i}^{yB}=$ ${y}_{j}{l}_{j,i}/{\sum }_{j=1}^{{M}_{i}^{B}}{y}_{j}={y}_{j}{l}_{j,i}/{Y}_{i}.$ The weights ${w}_{i}^{y}$ are directly given by (4.4), by replacing $x$ by $y.$ Estimator ${\stackrel{^}{Y}}_{y}$ obtained from (2.9) reduces to

${\stackrel{^}{Y}}_{y}=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{y}_{hj},$(4.7)

which is nothing else than estimator (3.1) obtained from the classical sampling theory.

Note that in general, this method requires one set of weighted links ${\theta }_{j,i}^{y}$ per variable of interest $y.$ One solution would be to restrict the determination of the weighted links to few key variables of interest, each associated with a larger set of correlated covariates. However, in the present situation, such a restriction is not necessary because estimator (4.7) corresponds simply to estimator (3.1). Indeed, at the end, we obtain estimation weights that simply correspond to the sampling weights.

For computing the variance of (4.7), we simply use formula (3.2). For the example of Section 3, we obtain $V\left({\stackrel{^}{Y}}_{y}\right)=V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480: this is a very large reduction compared to $V\left(\stackrel{^}{Y}\right)=$ 1,115,111.

## 4.2  Methods using weak-optimal weighted links

Method 4: Using weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{SRS}}$ under stratified SRSWoR

This method uses the weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{SRS}}$ of Deville and Lavallée (2006) described in Section 2.2. As mentioned earlier, these are obtained by minimising the variance (2.11) for a very specific choice of variable of interest: ${Y}_{i}=1$ for an enterprise $i$ of ${U}^{B}$ and ${Y}_{{i}^{\prime }}=0$ for all other enterprises ${i}^{\prime }$ of ${U}^{B}\left({i}^{\prime }\ne i\right).$ The resulting weak-optimal weighted links do not involve the variable $y,$ per se. Writing the values of ${\theta }_{j,i}^{w-\text{opt},\text{SRS}}$ involves expressions that can be cleverly expressed in matrix notation. Using summations, the expressions become much more complicated, because they involve a mixture of the joint selection probabilities ${\pi }_{j,{j}^{\prime }}^{A}$ of establishments $j$ and ${j}^{\prime }$ that can belong to the same stratum, or not.

Let us define the square matrix ${\Delta }^{A}=\left[{\Delta }_{j,{j}^{\prime }}^{A}\right]$ of size ${M}^{A}$ where ${\Delta }_{j,{j}^{\prime }}^{A}=$ $\left({\pi }_{j,{j}^{\prime }}^{A}-{\pi }_{j}^{A}{\pi }_{{j}^{\prime }}^{A}\right)/{\pi }_{j}^{A}{\pi }_{{j}^{\prime }}^{A}.$ Let ${\Gamma }^{A}=\left[{\gamma }_{j,{j}^{\prime }}^{A}\right]$ be the inverse of matrix ${\Delta }^{A},$ i.e., ${\left({\Delta }^{A}\right)}^{-1}={\Gamma }^{A}.$ Let ${\Gamma }_{i}^{A}=\left[{\gamma }_{ij,i{j}^{\prime }}\right]$ be the square submatrix of ${\Gamma }^{A}$ containing all elements (establishments) $\left(j,{j}^{\prime }\right)$ belonging to enterprise $i.$ Following Deville and Lavallée (2006), we have ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}=$ ${l}_{j,i}{\sum }_{{j}^{\prime }=1}^{{M}_{i}^{B}}{\gamma }_{ij,i{j}^{\prime }}/{\sum }_{j=1}^{{M}_{i}^{B}}{\sum }_{{j}^{\prime }=1}^{{M}_{i}^{B}}{\gamma }_{ij,i{j}^{\prime }}.$ Unfortunately, for the present case, the many-to-one correspondence between ${U}^{A}$ and ${U}^{B}$ does not help further in obtaining a simpler form for ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}.$

Note that if an enterprise $i$ contains an establishment ${j}_{0}$ in the take-all stratum $h=$ 1, we have ${\Delta }_{{j}_{0},{j}^{\prime }}^{A}=\left({\pi }_{{j}_{0},{j}^{\prime }}^{A}-{\pi }_{{j}_{0}}^{A}{\pi }_{{j}^{\prime }}^{A}\right)/{\pi }_{{j}_{0}}^{A}{\pi }_{{j}^{\prime }}^{A}=0$ and the matrix ${\Delta }^{A}$ is singular. In this case, the chosen solution is to set ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}=1$ for the take-all establishment ${j}_{0}$ of enterprise $i,$ and ${\stackrel{˜}{\theta }}_{{j}^{\prime },i}^{w-\text{opt},\text{SRS}}=0$ for the other establishments ${j}^{\prime }\ne {j}_{0}$ of enterprise $i.$ This means that ${Z}_{h{j}_{0}}^{w-\text{opt},\text{SRS}}=$ ${Y}_{i}$ and ${Z}_{h{j}^{\prime }}^{w-\text{opt},\text{SRS}}=0$ for ${j}^{\prime }\ne {j}_{0}:$ this means that the complete value ${Y}_{i}$ will be assigned to establishment ${j}_{0}$ that contributes 0 to the variance.

We have

${w}_{i}^{w-\text{opt},\text{SRS}}=\sum _{j=1}^{{M}_{i}^{B}}\frac{{t}_{j}^{A}}{{\pi }_{j}^{A}}{\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}.$(4.8)

Using (4.8), we can rewrite estimator (2.9) as

${\stackrel{^}{Y}}_{w-\text{opt},\text{SRS}}=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{Z}_{hj}^{w-\text{opt},\text{SRS}}$(4.9)

where

${Z}_{hj}^{w-\text{opt},\text{SRS}}=\sum _{i=1}^{{N}^{B}}{Y}_{i}{\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}$(4.10)

for $j\in h$ and $j\in i.$

To compute the variance of ${\stackrel{^}{Y}}_{w-\text{opt},\text{SRS}},$ we use formula (2.11) with the values (4.10). For the example of Section 3, we get ${V\left(\stackrel{^}{Y}}_{w-\text{opt},\text{SRS}}\right)=$ 23,111, which is a tremendous reduction of variance compared to both $V\left(\stackrel{^}{Y}\right)=$ 1,115,111 and $V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480.

Method 5: Using weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{PS}}$ under Poisson Sampling

In the context of business surveys, Poisson Sampling selects sample ${s}^{A}$ by going through the ${M}^{A}$ establishments of population ${U}^{A}$ and selecting establishment $j$ if ${u}_{j}\le {\pi }_{j}^{A},$ where ${u}_{j}\sim U\left(0,1\right).$ The selection probabilities are simply given by ${\pi }_{j}^{A}={m}_{h}^{A}/{M}_{h}^{A}$ for $j\in h$ and the resulting realised stratum sample size ${\stackrel{˜}{m}}_{h}^{A}$ is random. In this context, this sampling design can also be seen as stratified Bernoulli Sampling (see Särndal, Swensson and Wretman 1992).

Poisson Sampling (or stratified Bernoulli Sampling) is a very simple sample design. As it can be noted, the selection of each establishment of ${s}^{A}$ is done independently from one establishment to another. This means that the joint selection probability ${\pi }_{j,{j}^{\prime }}^{A}$ of two different establishments $j$ and ${j}^{\prime }$ of ${U}^{A}$ is simply given by ${\pi }_{j,{j}^{\prime }}^{A}={\pi }_{j}^{A}{\pi }_{{j}^{\prime }}^{A}.$ By conditioning on ${\stackrel{˜}{m}}_{h}^{A},$ it can be shown that stratified Bernoulli Sampling corresponds to stratified SRSWoR. The estimator to be used with stratified Bernoulli Sampling is the ratio estimator

${\stackrel{˜}{Y}}_{\theta }=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{\stackrel{˜}{m}}_{h}^{A}}\sum _{j=1}^{{\stackrel{˜}{m}}_{h}^{A}}{Z}_{hj}^{\theta }.$(4.11)

The variance of estimator (4.11) is approximately given by formula (2.11) (see Brewer and Hanif 1983). Because of the relative closeness between the two designs, assuming Poisson Sampling can be a reasonable approach for computing the weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{PS}}.$

The weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{PS}}$ are obtained by computing ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{SRS}}$ as in Method 4, but assuming that sample selection is done using Poisson Sampling. This assumption significantly simplifies the calculations because the matrix ${\Delta }^{A}$ then becomes a diagonal matrix, which is easy to invert. Because of the many-to-one correspondence between ${U}^{A}$ and ${U}^{B},$ we obtain after the minimisation process

${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{PS}}=\frac{{\pi }_{j}^{A}{l}_{j,i}}{\left(1-{\pi }_{j}^{A}\right){\tau }_{i}}$(4.12)

where ${\tau }_{i}={\sum }_{j=1}^{{M}_{i}^{B}}{\pi }_{j}^{A}/\left(1-{\pi }_{j}^{A}\right).$ Therefore, from (2.8), we have

${w}_{i}^{w-\text{opt},\text{PS}}=\frac{1}{{\tau }_{i}}\sum _{j=1}^{{M}_{i}^{B}}\frac{{t}_{j}^{A}}{\left(1-{\pi }_{j}^{A}\right)}.$(4.13)

Using (4.13), we can rewrite (2.9) as

${\stackrel{^}{Y}}_{w-\text{opt},\text{PS}}=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{Z}_{hj}^{w-\text{opt},\text{PS}}$(4.14)

where

${Z}_{hj}^{w-\text{opt},\text{PS}}=\frac{{\pi }_{j}^{A}}{\left(1-{\pi }_{j}^{A}\right)}\frac{{Y}_{i}}{{\tau }_{i}}$(4.15)

for $j\in h$ and $j\in i.$ Note that the previous results assume that $0<{\pi }_{j}^{A}<1$ for all establishments $j$ of ${U}^{A}.$ For the case where ${\pi }_{j}^{A}=1$ for a given establishment ${j}_{0}$ of an enterprise $i,$ we set ${\stackrel{˜}{\theta }}_{{j}_{0},i}^{w-\text{opt},\text{PS}}=1,$ and ${\stackrel{˜}{\theta }}_{{j}^{\prime },i}^{w-\text{opt},\text{PS}}=0$ for ${j}^{\prime }\ne {j}_{0}.$ For computing the variance of ${\stackrel{^}{Y}}_{w-\text{opt},\text{PS}},$ we use formula (2.11) with the $Z­$ values given by (4.15).

For the example of Section 3, we get ${V\left(\stackrel{^}{Y}}_{w-\text{opt},\text{PS}}\right)=$ 22,857. Again, this is a very large reduction of variance compared to both $V\left(\stackrel{^}{Y}\right)=$ 1,115,111 and $V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480.

Method 6: Using weak-optimal weighted links ${\theta }_{j,i}^{w-\text{opt},\text{grp}}$ under Poisson Sampling of grouped-establishments

This method consists once more in using the weak-optimal weighted links of Deville and Lavallée (2006) described in Section 2.2, but with grouped-establishments. As a first step, we build grouped-establishments in the population ${U}^{A}$ where a grouped-establishment ${j}^{*}$ consists in all establishments that are part of the same stratum $h$ and that are belonging to the same enterprise $i.$ This creates a new population ${U}^{{A}^{*}}$ containing ${M}^{{A}^{*}}$ grouped-establishments. The sample ${s}^{{A}^{*}}$ of ${m}^{{A}^{*}}$ grouped-establishments contains all grouped-establishments formed from the establishments of sample ${s}^{A}.$ The selection probability of the grouped-establishment ${j}^{*}$ is given by

${\pi }_{{j}^{*}}^{{A}^{*}}=1-\frac{\left(\begin{array}{c}{M}_{h}^{A}-{M}_{{j}^{*}}\\ {m}_{h}^{A}\end{array}\right)}{\left(\begin{array}{c}{M}_{h}^{A}\\ {m}_{h}^{A}\end{array}\right)}=1-\frac{\left({M}_{h}^{A}-{M}_{{j}^{*}}\right)\left({M}_{h}^{A}-{M}_{{j}^{*}}-1\right)...\left({M}_{h}^{A}-{M}_{{j}^{*}}-{m}_{h}^{A}+1\right)}{{M}_{h}^{A}\left({M}_{h}^{A}-1\right)...\left({M}_{h}^{A}-{m}_{h}^{A}+1\right)}$(4.16)

for ${j}^{*}\in h,$ where ${M}_{{j}^{*}}$ is the number of establishments within the grouped-establishment ${j}^{*}.$

The rationale behind the use of grouped-establishments is to have only one unit per stratum belonging to a given enterprise. Because, by construction, the grouped-establishments ${j}^{*}$ of an enterprise $i$ belong to different strata, their selection is done independently from one grouped-establishment to another. This implies that the solution to weak optimality is then similar to the one obtained in Section 4.5 for Poisson Sampling, but with grouped-establishments. Therefore, we have

${\stackrel{˜}{\theta }}_{{j}^{*},i}^{w-\text{opt},\text{grp}}=\frac{{\pi }_{{j}^{*}}^{A}{l}_{{j}^{*},i}}{\left(1-{\pi }_{{j}^{*}}^{A}\right){\tau }_{i}^{*}}$(4.17)

where ${\tau }_{i}^{*}={\sum }_{{j}^{*}=1}^{{M}_{i}^{{B}^{*}}}{\pi }_{{j}^{*}}^{{A}^{*}}/\left(1-{\pi }_{{j}^{*}}^{{A}^{*}}\right)$ and ${M}_{i}^{{B}^{*}}$ is the number of groups-establishments contained in enterprise $i.$

The use of grouped-establishments can be seen as an intermediate step in the Indirect Sampling process going from population ${U}^{A}$ to population ${U}^{B}.$ That is, the Indirect Sampling process goes from population ${U}^{A}$ to population ${U}^{{A}^{*}},$ and then from population ${U}^{{A}^{*}}$ to population ${U}^{B}.$ In the present case, we have $j\in {j}^{*}\in i$ for all establishments. Following the rules of transitivity defined by Deville and Lavallée (2006), we can show that the weak-optimal weighted links ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{grp}}$ for $j\in {j}^{*}$ and ${j}^{*}\in i$ (and thus, $j\in i\right)$ are given by

${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{grp}}=\frac{{\pi }_{{j}^{*}}^{A}{l}_{j,i}}{\left(1-{\pi }_{{j}^{*}}^{A}\right){\tau }_{i}^{*}{M}_{{j}^{*}}}.$(4.18)

Therefore, from (2.8), we have

${w}_{i}^{w-\text{opt},\text{grp}}=\frac{1}{{\tau }_{i}^{*}}\sum _{{j}^{*}=1}^{{M}_{i}^{{B}^{*}}}\frac{{\pi }_{{j}^{*}}^{{A}^{*}}}{\left(1-{\pi }_{{j}^{*}}^{{A}^{*}}\right){M}_{{j}^{*}}}\sum _{j=1}^{{M}_{{j}^{*}}}\frac{{t}_{j}^{A}}{{\pi }_{j}^{A}}.$(4.19)

Using (4.19), we can rewrite (2.9) as

${\stackrel{^}{Y}}_{w-\text{opt},\text{grp}}=\sum _{h=1}^{H}\frac{{M}_{h}^{A}}{{m}_{h}^{A}}\sum _{j=1}^{{m}_{h}^{A}}{Z}_{hj}^{w-\text{opt},\text{grp}}$(4.20)

where

${Z}_{hj}^{w-\text{opt},\text{grp}}=\frac{{\pi }_{{j}^{*}}^{{A}^{*}}}{\left(1-{\pi }_{{j}^{*}}^{{A}^{*}}\right){M}_{{j}^{*}}}\frac{{Y}_{i}}{{\tau }_{i}^{*}}$(4.21)

for $j\in h$ and ${j}^{*}\in h.$ Note that the previous results assume that $0<{\pi }_{{j}^{*}}^{{A}^{*}}<1$ for all grouped-establishments ${j}^{*}$ of ${U}^{{A}^{*}}.$ For the case where ${\pi }_{{j}^{*}}^{{A}^{*}}=1$ for a given grouped-establishment ${j}_{0}^{*}$ of an enterprise $i,$ we set ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{grp}}={l}_{j,i}/{M}_{{j}^{*}}$ for the all the establishments $j$ of this grouped-establishment ${j}_{0}^{*},$ and ${\stackrel{˜}{\theta }}_{j,i}^{w-\text{opt},\text{grp}}=0$ for all other establishments not part of the grouped-establishment ${j}_{0}^{*}.$ We have ${\pi }_{{j}^{*}}^{{A}^{*}}=1$ when at least one establishment $j$ belonging to ${j}^{*}$ has ${\pi }_{j}^{A}=1.$ For computing the variance of ${\stackrel{^}{Y}}_{w-\text{opt},\text{grp}},$ we use formula (2.11) with the values (4.21).

For the example of Section 3, we get ${V\left(\stackrel{^}{Y}}_{w-\text{opt},\text{grp}}\right)=$ 23,000. Again, this is a very significant reduction of variance compared to both $V\left(\stackrel{^}{Y}\right)=$ 1,115,111 and $V\left({\stackrel{^}{Y}}_{\text{classic}}\right)=$ 80,480.

## 4.3  Other methods

Method 7: Using a designated establishment

As mentioned before, the rationale behind the use of grouped-establishments in Method 6 is to have only one unit per stratum belonging to a given enterprise. Using a similar idea, one can decide on a single establishment that will represent the complete enterprise. That is, for each enterprise belonging to ${U}^{B},$ we identify one establishment of ${U}^{A}$ that will be used for the selection of its owning enterprise. A natural choice for the designated establishment within the enterprise is the one with the largest value for a given variable $x.$ For example, $x$ can be the establishment's revenue.

Choosing a designated establishment yields a new sampling frame ${U}^{{A}_{+}}$ that contains the same number of units as the target population ${U}^{B},$ i.e., ${M}^{{A}_{+}}={N}^{B}.$ Since there is a one-to-one correspondence between the designated establishment and its owning enterprise, the designated establishment of enterprise $i$ may also be labelled using $i.$ The new frame ${U}^{{A}_{+}}$ can keep the same stratification definition as the original frame ${U}^{A}.$ That is, if the stratification of ${U}^{A}$ was done by province and industrial classes based on the establishments' values, the stratification of ${U}^{{A}_{+}}$ is done by the same classes based on the designated establishments' values.

From the sampling frame ${U}^{{A}_{+}},$ we select a sample ${s}^{{A}_{+}}$ of ${m}^{{A}_{+}}$ designated establishments with stratified SRSWoR by using sampling fractions equal to the original ones, i.e., ${\pi }_{i}^{{A}_{+}}={m}_{h}^{{A}_{+}}/{M}_{h}^{{A}_{+}}={m}_{h}^{A}/{M}_{h}^{A},$ for $i\in h.$ The estimation of the total $Y$ is obtained using the following estimator:

${\stackrel{^}{Y}}_{+}=\sum _{h=1}^{H}\frac{{M}_{h}^{{A}_{+}}}{{m}_{h}^{{A}_{+}}}\sum _{i=1}^{{m}_{h}^{{A}_{+}}}{Y}_{i}.$(4.22)

It can be shown that estimator (4.22) is unbiased, and its variance is given by

$\text{Var}\left({\stackrel{^}{Y}}_{+}\right)=\sum _{h=1}^{H}{M}_{h}^{{A}_{+}}\left(\frac{{M}_{h}^{{A}_{+}}-{m}_{h}^{{A}_{+}}}{{m}_{h}^{{A}_{+}}}\right){S}_{+Yh}^{2}$(4.23)

where ${S}_{+Yh}^{2}=\sum _{i=1}^{{M}_{h}^{{A}_{+}}}{\left({Y}_{hi}-{\overline{Y}}_{h}\right)}^{2}/\left({M}_{h}^{{A}_{+}}-1\right)$ and ${\overline{Y}}_{h}=\sum _{i=1}^{{M}_{h}^{{A}_{+}}}{Y}_{hi}/{M}_{h}^{{A}_{+}}.$

Note that although we only keep one designated establishment per enterprise, we are still able to produce estimates per stratum, or for any domain of interest (e.g., different industrial activities). For example, let us consider the small example of Section 3. For the first enterprise of ${U}^{B}$ (i.e., the one with a total revenue of 2,400), the designated establishment would be the first establishment of the take-all stratum of ${U}^{A}$ (i.e., the establishment with 25 employees). None of the three other establishments of this enterprise would be available for sampling. However, if we were interested in producing an estimate for the second stratum, we would simply restrict the computation of the values ${Y}_{i}$ in (4.22) to the establishments belonging to this second stratum. In the present case, rather than using ${Y}_{i}=$ 2,400 in (4.22), we would then use ${Y}_{i}=$ 300. This corresponds to domain estimation (Särndal, et al. 1992).

For the example of Section 3, we obtain $V\left({\stackrel{^}{Y}}_{+}\right)=$ 1,820,000! With this method, since an establishment inherits all revenues of the enterprise, the use of a designated establishment is advantageous when this establishment is in the take-all stratum. However, the designated establishment may itself be contained in a take-some stratum, and this results in a stratum that is even more skewed. The total revenue of the enterprise, multiplied by the sampling weight, is assigned to this take-some stratum, and this increases the variance significantly.

Method 8: Using the selection probabilities of the enterprises

As mentioned in Lavallée (2002, 2007), using the Rao-Blackwell theorem, sufficient statistics can improve an existing estimator by producing a new estimator with a mean squared error that is smaller than or equal to that of the initial estimator (see Cassel, Särndal and Wretman 1977). Note that this form of improvement was used, for instance, by Thompson (1990) in the context of Adaptive Cluster Sampling.

Starting from estimator (2.1) (or (2.3)), the estimator ${\stackrel{^}{Y}}_{\text{RB}}$ obtained by using the Rao-Blackwell theorem is given by

${\stackrel{^}{Y}}_{\text{RB}}=\sum _{i=1}^{{n}^{B}}\frac{{Y}_{i}}{{L}_{i}^{B}}\sum _{j=1}^{{M}^{A}}\frac{P\left({t}_{j}^{A}=1|{s}^{B}\right)}{{\pi }_{j}^{A}}{l}_{j,i}$(4.24)

where $P\left({t}_{j}^{A}=1|{s}^{B}\right)$ is the probability of having selected establishment $j$ from ${U}^{A},$ given that the ${n}^{B}$ enterprises of ${s}^{B}$ have been selected from ${U}^{B}.$

Using the many-to-one correspondence between ${U}^{A}$ and ${U}^{B},$ an approximation to the probability $P\left({t}_{j}^{A}=1|{s}^{B}\right)$ can be obtained. That is, for $j\in i,$ we have

$P\left({t}_{j}^{A}=1|{s}^{B}\right)\approx P\left({t}_{j}^{A}=1|i\in {s}^{B}\right)$
$=P\left({t}_{j}^{A}=1,i\in {s}^{B}\right)/P\left(i\in {s}^{B}\right)$(4.25)
$=P\left({t}_{j}^{A}=1\right)/P\left(i\in {s}^{B}\right)$
$={\pi }_{j}^{A}/{\pi }_{i}^{B}$

where ${\pi }_{i}^{B}$ is the selection probability of enterprise $i\in {U}^{B},$ which corresponds to the probability of selecting any of its ${M}_{i}^{B}$ establishments. Note that result (4.25) becomes exact in the context of Poisson Sampling. Using (4.25), estimator (4.24) is then approximately equivalent to the following Horvitz-Thompson estimator

${\stackrel{^}{Y}}_{\text{HT}}=\sum _{i=1}^{{n}^{B}}\frac{{Y}_{i}}{{\pi }_{i}^{B}}.$(4.26)

Since estimator (4.26) is nothing else than a Horvitz-Thompson estimator based on the selection of enterprises, its variance is given by

${\stackrel{^}{Y}}_{\text{HT}}=\sum _{i=1}^{{N}^{B}}\sum _{{i}^{\prime }=1}^{{N}^{B}}\frac{\left({\pi }_{i,{i}^{\prime }}^{B}-{\pi }_{i}^{B}{\pi }_{{i}^{\prime }}^{B}\right)}{{\pi }_{i}^{B}{\pi }_{{i}^{\prime }}^{B}}{Y}_{i}{Y}_{{i}^{\prime }}.$(4.27)

The computation of the selection probability ${\pi }_{i}^{B}$ requires the knowledge of the selection probabilities ${\pi }_{j}^{A}$ of all ${M}_{i}^{B}$ establishments of enterprise $i.$ In general, this can be difficult or even impossible to obtain (see Lavallée 2002, 2007). This can be a severe barrier for using estimator (4.26) in practice, and actually, this is one of the driving reason why using the GWSM. However, in the present case, this reveals to be possible because the complete frame ${U}^{A}$ is available for the selection of the establishments. The task is also simplified by the use of stratified SRSWoR. The selection probabilities ${\pi }_{i}^{B}$ can then be computed by adapting formula (4.16). It is also possible to compute the joint selection probabilities ${\pi }_{i,{i}^{\prime }}^{B},$ but this is more difficult.

For the example of Section 3, we obtain $V\left({\stackrel{^}{Y}}_{\text{HT}}\right)=$ 14,545, and this value corresponds to the lowest variance of the proposed methods.

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