13 Indirect Sampling

In some situations, the statistician has no sampling frame available… This absence implies that it is not possible to define the selection probability of a sample, making it impossible to calculate the sampling error. In other situations, the statistician does have access to sampling frames that do not correspond directly to the sampling units of the target population. Lavallé (2007)

In practice, element sampling frames are not always available. However, in some cases it is possible to access different list frames of elements that, while not part of the target population, are indirectly related to it. The process of selecting samples under these conditions is called indirect sampling, and it is characterized by the fact that producing estimates of simple totals or means can become a nightmare for the statistician. To solve this problem, the generalized weighting method is used. It is characterized by its simplicity, and its estimators share the desirable property of unbiasedness, even under indirect sampling.

To produce estimates in social, economic, and similar research generated through a sampling strategy $(p(s),\hat{T}(S))$, access, at least implicitly, to an element sampling frame of the target population, denoted by $U_B$, is essential. Unfortunately, access to such a sampling frame is often difficult to obtain. However, sometimes it is possible to consider the availability of an element sampling frame¹ from some other population $U_A$ linked to the elements of the target population.

The goal is to select a probability sample $s_A$ from population $U_A$ to obtain estimates for population $U_B$ using the correspondence between the two populations. For example, assume that estimates are desired for a population of children, with the restriction that only a list of parents, containing the identification and location of each one, is available. The target population is the children, but it is necessary to select a sample of parents in order to interview the children.

13.1 Notation

Population $U_A$ contains $N_A$ units. Each unit belonging to population $U_A$ will be labeled with the letter $j$. Each unit belonging to the target population $U_B$ of size $N_B$ will be labeled with the letter $i$. The correspondence between the two populations $U_A$ and $U_B$ can be represented by a link matrix, denoted by $\Theta_{AB}=[\theta_{ij}^{AB}]$, of size $N_A\times N_B$. The possible values of the matrix are given as follows: \[ \theta_{ij}^{AB}\left\{ \begin{array}{ll} >0, & \hbox{if $j$ is related to $i$;} \\ =0, & \hbox{otherwise.} \end{array} \right. \tag{13.1}\]

In the parents example, if the link matrix is given by \[ \Theta_{AB}= \left( \begin{array}{ccc} \theta_{11}^{AB} & \theta_{12}^{AB} & 0 \\ 0 & \theta_{22}^{AB} & 0 \\ 0 & 0 & \theta_{33}^{AB} \\ 0 & 0 & \theta_{43}^{AB} \\ \end{array} \right) \]

then the existing links between the two populations would be the following:

The first couple, given by elements 1 and 2 of population $U_A$, has one child, recorded as the second element of population $U_B$.
However, element 1 of population $U_A$ has another child outside the marriage, recorded as the first element of population $B$.
The second couple, given by elements 3 and 4 of population $U_A$, has only one child, recorded as the third and last element of population $U_B$.

Usually, when there is a link between the $j$-th element of population $U_A$ and the $i$-th element of population $U_B$, $\theta_{ij}^{AB}$ takes the value one. However, the link can be different from one, as discussed in (Lavallé 2007).

Using indirect sampling, a sample $s_A$ of size $n_A$ is selected through a sampling design $p_A(s_A)$. Let $\pi^A_j>0$ $\forall j \in U_A$ be the inclusion probability of the $j$-th element. For each element in the sample $s_A$, the units in $U_B$ whose correspondence with the elements of population $U_A$ is nonzero are identified, that is, those such that $\theta_{ij}^{AB}>0$. Let $s_B$ be the set of $n_B$ target population units that could be identified with the help of the elements belonging to population $U_A$. Therefore

\[ s_B=\{i\in B \mid \exists j\in s_A\text{ and }\theta_{ij}^{AB}> 0 \} \]

For each identified element in the target population, the characteristic of interest $y$ is measured. However, the number of target population elements identified by the indirect sampling process is generally random because it depends not only on the selected sample $s_A$, but also on the link matrix $\Theta_{AB}$. Therefore, it becomes very complicated to establish a budget for the information collection stage. Fortunately, in some populations, such as parents and children, it is possible to predict the number of links between the populations; for example, a parent has one, two, or even three children.

An important requirement when applying indirect sampling is that, for all units selected in the sample $s_A$, the correspondence to the target population can be obtained, and vice versa. This is a very strong, though necessary, assumption. For example, it is easy for a parent to identify all their children; on the other hand, it is not so easy for a very young child to identify divorced parents. However, this operational problem is considered negligible in terms of the theoretical development. Thus, it is possible to know the values of the matrix $\Theta_{AB}$ for the rows $j\in s_A$ as well as for the columns $i\in s_B$.

13.2 Estimation of the Total

The objective is to estimate the total of $y$ in the target population: \[ \begin{aligned} t_y&=\sum_{i\in U_B}y_i\\ &=\mathbf{1}_B\mathbf{y} \end{aligned} \]

where $\mathbf{1}_B$ is the vector of ones of size $N_B$ and $\mathbf{y}=(y_1,\ldots,y_{N_B})'$. Now, the following definition will assist in estimating the population total.

Definition

The standard link matrix is defined as \[ \begin{aligned} \tilde{\Theta}_{AB}=\Theta_{AB}[\textrm{diag}(\mathbf{1}_A\Theta_{AB})]^{-1} \end{aligned} \] Based on the above, note that \[ \begin{aligned} \mathbf{1}_A'\Theta_{AB}=(\theta_{+1}^{AB},\theta_{+2}^{AB},\ldots,\theta_{+N_B}^{AB}) \end{aligned} \]

where $\theta_{+i}^{AB}=\sum_{j\in U_A}\theta_{ji}^{AB}$ must be nonzero² for all $i\in U_B$. With this, (14.2.3) is well defined and therefore $\tilde{\Theta}_{ji}^{AB}=\dfrac{\theta_{ji}^{AB}}{\theta_{+i}^{AB}}$.

In the example population, this would mean that every child must be linked to at least one parent, which is logical in this specific context. However, this logic does not always hold, and in some cases the definition of population $U_A$ itself is complex.

Result

If $\tilde{\Theta}_{AB}$ is a standard link matrix, then \[ \begin{aligned} \tilde{\Theta}_{AB}'\mathbf{1}_A=\mathbf{1}_B \end{aligned} \]

Proof. Expanding algebraically, the proof follows directly by applying the previous definition as follows: \[ \begin{aligned} \tilde{\Theta}_{AB}'\mathbf{1}_A&=\left(\left[\textrm{diag}(\mathbf{1}_A\Theta_{AB})\right]^{-1}\right)'\Theta_{AB}'\mathbf{1}_A\\ &=[\textrm{diag}(\mathbf{1}_A\Theta_{AB})]^{-1}(\mathbf{1}_A'\Theta_{AB})'\\ &=\left( \begin{array}{cccc} \frac{1}{\theta_{+1}^{AB}} & 0 & \cdots & 0 \\ 0 & \frac{1}{\theta_{+2}^{AB}} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & \frac{1}{\theta_{+N_B}^{AB}} \end{array} \right)\left( \begin{array}{c} \theta_{+1}^{AB} \\ \theta_{+2}^{AB} \\ \vdots \\ \theta_{+N_B}^{AB} \\ \end{array} \right)=\mathbf{1}_B \end{aligned} \]

Result

The population total of the characteristic of interest can be rewritten as follows: \[ \begin{aligned} t_y=\sum_{j \in U_A}\sum_{i \in U_B}\dfrac{\theta_{ji}^{AB}}{\theta_{+i}^{AB}}y_i \end{aligned} \]

Proof. Directly from the definition of the standard link matrix, we have \[ \begin{aligned} t_y&=\mathbf{1}_B'\mathbf{y}\\ &=\mathbf{1}_A'\tilde{\Theta}_{AB}\mathbf{y}=\sum_{j \in U_A}\sum_{i \in U_B}\dfrac{\theta_{ji}^{AB}}{\theta_{+i}^{AB}}y_i \end{aligned} \]

Next, define the column vector $\mathbf{z}=\tilde{\Theta}_{AB}\mathbf{y}$ of size $N_A$, whose $j$-th element is $z_j=\sum_{i \in U_B}\tilde{\theta}_{ji}^{AB}y_i$, established for population $U_A$ and measured in the sample $s_A$. To estimate $t_y$, the values of $y_i$ measured in the sample $s_B$ must be used, so that the following estimator can be constructed: \[ \begin{aligned} \hat{t}_y&=\sum_{i\in U_B}w_iy_i\\ &=\mathbf{w}'\mathbf{y} \end{aligned} \]

where $\mathbf{w}=(w_1,\ldots,w_{N_B})$, and $w_i$ is the estimated weight of the $i$-th element of $s_B$. Of course, $w_i=0$ if $i\notin s_B$. For $\hat{t}_y$ to be unbiased, it is usual to define $w_i=(\pi_i^B)^{-1}$. However, this choice, although possible in theory, is very difficult to find in indirect sampling because all possible links generated by all possible samples $a_A$ must be known.

13.3 Generalized Weighting Method

The sample $s_A$ was selected according to a sampling design $p_A(s_A)$. This sampling design induces a vector of inclusion probabilities for all elements of $U_A$. Let $\mathbf{\boldsymbol{\Pi}}_A=diag(\pi^A_1,\ldots,\pi^A_{N_A})$ be a diagonal matrix of size $N_A\times N_A$ containing the inclusion probabilities for $j\in U_A$ on its diagonal. Similarly, define the inclusion matrix of the elements in the sample, given by $\mathbf{I_A}=diag(I_1^A,\ldots,I_{N_A}^A)$, with \[ I_j^A(S_A)= \begin{cases} 1 & \text{if $i \in S_A$}\\ 0 & \text{if $i \notin S_A$}. \end{cases} \]

Starting from the fact that the population total takes the following form, \[ \begin{aligned} t_y&=\mathbf{1}_A'\tilde{\Theta}_{AB}\mathbf{y}\\ &=\mathbf{1}_A'\mathbf{z} \end{aligned} \]

it is possible to construct an expression that respects the principles of the Horvitz-Thompson estimator in terms of the vector $\mathbf{Z}$. Therefore, \[ \begin{aligned} \hat{t}_y=\hat{t}_{z,\pi}&=\mathbf{1}_A'\mathbf{I_A}\mathbf{\boldsymbol{\Pi}}_A^{-1}\mathbf{z}\\ &=\mathbf{1}_A'\mathbf{I_A}\mathbf{\boldsymbol{\Pi}}_A^{-1}\tilde{\Theta}_{AB}\mathbf{y} \end{aligned} \]

Thus, the vector of weights for the target population $U_B$ is defined as \[ \begin{aligned} \mathbf{w}=\mathbf{1}_A'\mathbf{I_A}\mathbf{\boldsymbol{\Pi}}_A^{-1}\tilde{\Theta}_{AB} \end{aligned} \]

where each element of $\mathbf{w}$, which is a vector of size $N_B$, is defined by the following expression: \[ \begin{aligned} w_i= \begin{cases} \sum_{j\in U_A}I_j\dfrac{\tilde{\Theta}_{ji}^{AB}}{\pi_j^A}, & \text{for all $i\in s_B$}\\ 0, & \text{for all $i\notin s_B$} \end{cases} \end{aligned} \]

In this way, the weights $w_i$ are said to have been obtained through the generalized weighting method, as described in (Lavallé 2007). Along these lines, returning to our example of the population of parents and children, if the realized sample of parents were given by

then the set of children identified by the selected parents would be given by

and the resulting weights would be

$w_1=0$ because Child 1 was not identified by any parent.
For Child 2, we have \[ \begin{aligned} w_2&=\sum_{j\in U_A}I_j\dfrac{\tilde{\Theta}_{j2}^{AB}}{\pi_1^A}\\ &=\sum_{j\in s_A}\dfrac{\tilde{\Theta}_{j2}^{AB}}{\pi_1^A}\\ &=\dfrac{\tilde{\Theta}_{22}^{AB}}{\pi_2^A}+\dfrac{\tilde{\Theta}_{23}^{AB}}{\pi_3^A}\\ &=\dfrac{\Theta_{22}^{AB}}{\Theta_{+2}^{AB}}\dfrac{1}{\pi_2^A}+\dfrac{\Theta_{23}^{AB}}{\Theta_{+2}^{AB}}\dfrac{1}{\pi_3^A} \end{aligned} \]
For Child 3, we have \[ \begin{aligned} w_3&=\sum_{j\in U_A}I_j\dfrac{\tilde{\Theta}_{j3}^{AB}}{\pi_1^A}\\ &=\sum_{j\in s_A}\dfrac{\tilde{\Theta}_{j3}^{AB}}{\pi_1^A}\\ &=\dfrac{\tilde{\Theta}_{32}^{AB}}{\pi_2^A}+\dfrac{\tilde{\Theta}_{33}^{AB}}{\pi_3^A}\\ &=\dfrac{\Theta_{32}^{AB}}{\Theta_{+3}^{AB}}\dfrac{1}{\pi_2^A}+\dfrac{\Theta_{33}^{AB}}{\Theta_{+3}^{AB}}\dfrac{1}{\pi_3^A} \end{aligned} \]

13.3.1 Properties

The following properties are generated by the weights from the generalized weighting method.

Result

The estimator $t_y$ is unbiased.

Proof. It is enough to show that $E(\mathbf{w})=\mathbf{1}_B$. This holds by construction, since the Horvitz-Thompson estimator is unbiased because the expectation of the indicator variables $I_j$ is equal to the inclusion probability $\pi_j^A$. Thus, we have \[ \begin{aligned} E(\hat{t}_y)&=E(\mathbf{w})y\\ &=E(\mathbf{1}_A'\mathbf{I_A}\mathbf{\boldsymbol{\Pi}}_A^{-1}\tilde{\Theta}_{AB})y\\ &=\mathbf{1}_A'E(\mathbf{I_A})\mathbf{\boldsymbol{\Pi}}_A^{-1}\tilde{\Theta}_{AB}y\\ &=\mathbf{1}_A'\mathbf{\boldsymbol{\Pi}}_A\mathbf{\boldsymbol{\Pi}}_A^{-1}\tilde{\Theta}_{AB}y\\ &=\mathbf{1}_A'\tilde{\Theta}_{AB}y\\ &=\mathbf{1}_B'y=t_y \end{aligned} \]

Result

The vector $\mathbf{w}$ provides unbiased estimates if and only if the matrix $\tilde{\Theta}_{AB}$ is a standard link matrix.

Proof. We have $E(\mathbf{w})=\tilde{\Theta}_{AB}'\mathbf{1}_A$; however, assuming from the previous result that the weight vector induces unbiased estimates, we have $E(\mathbf{w})=\mathbf{1}_B$. Therefore, $\tilde{\Theta}_{AB}'\mathbf{1}_A=\mathbf{1}_B$, and this reasoning proves the result.

Result

The variance of $\hat{t}_y$ is given by \[ \begin{aligned} Var(\hat{t}_y)&=\mathbf{z}'\mathbf{\Delta}_A\mathbf{z}\\ &=\mathbf{y}'\mathbf{\Delta}_{B}\mathbf{y} \end{aligned} \] where $\mathbf{\Delta}_{B}=\tilde{\Theta}_{AB}'\mathbf{\Delta}_{A}\tilde{\Theta}_{AB}$, and $\mathbf{\Delta}_A$ is the variance-covariance matrix of size $N_A\times N_A$ of the indicator variables of the elements of population $U_A$, doubly weighted by inclusion probabilities, whose $jj'$ element is given by \[ [\mathbf{\Delta}_A]_{jj'}=\frac{\Delta_{jj'}^A}{\pi_j^A\pi_{j'}^A}=\frac{\pi_{jj'}^A-\pi_{j}^A\pi_{j'}^A}{\pi_j^A\pi_{j'}^A} \]

Proof. Following the principles of the Horvitz-Thompson estimator, the proof is immediate, since

\[ \begin{aligned} \mathbf{z}'\mathbf{\Delta}_A\mathbf{z}&=(z_1,\ldots,z_{N_A}) \begin{pmatrix} \frac{\Delta_{11}^A}{\pi_1^A\pi_{1}^A} & \ldots & \frac{\Delta_{1N_A}^A}{\pi_1^A\pi_{N_A}^A} \\ \vdots & \ddots & \vdots \\ \frac{\Delta_{N_A1}^A}{\pi_{N_A}^A\pi_1^A} & \ldots & \frac{\Delta_{N_AN_A}^A}{\pi_{N_A}^A\pi_{N_A}^A} \\ \end{pmatrix} \begin{pmatrix} z_1 \\ \vdots \\ z_{N_A} \\ \end{pmatrix}\\ &=\left(\sum_{j\in U_A}z_j\frac{\Delta_{j1}^A}{\pi_1^A\pi_{j}^A},\ldots,\sum_{j\in U_A}z_j\frac{\Delta_{jN_A}^A}{\pi_{N_A}^A\pi_{j}^A}\right) \begin{pmatrix} z_1 \\ \vdots \\ z_{N_A} \\ \end{pmatrix}\\ &=\sum_{j\in U_A}\sum_{j'\in U_A}\Delta_{jj'}\frac{z_j}{\pi_j^A}\frac{z_{j'}}{\pi_{j'}^A}=Var(\hat{t}_{z,\pi}) \end{aligned} \] and replacing conveniently gives the proof.

13.3.2 Some Special Matrices

In general, indirect sampling produces unbiased estimates if the generalized weighting method is used. However, it is worth presenting special cases of link matrices that illustrate the behavior of the Horvitz-Thompson estimator. This subsection presents some of these matrices, which correspond to extreme cases. Although they may not be plausible in practice, they serve to illustrate the effect of the link matrix on the estimator of the population total of the characteristic of interest.

13.3.2.1 Identity Matrix

Assuming that the link matrix is an identity matrix, population $U_A$ and population $U_B$ have a one-to-one correspondence. This implies that the size of the two populations is the same, so $N_A=N_B=N$, and that the link matrix is given by \[ \begin{aligned} \Theta_{AB}=\mathbf{I}_{N\times N}= \left( \begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 1 \end{array} \right) \end{aligned} \]

Thus, the vector of weights is \[ \mathbf{w}=\left(\frac{I^A_1}{\pi_1^A},\cdots,\frac{I^A_{N_A}}{\pi_{N_A}^A}\right)' \]

and therefore $\mathbf{Z=y}$. Thus, the estimator $\hat{t}_y$ takes the form of the Narain-Horvitz-Thompson estimator as follows: \[ \begin{aligned} \hat{t}_y=\hat{t}_{y,N\pi}&=\mathbf{1}_A'\mathbf{I_A}\mathbf{\boldsymbol{\Pi}}_A^{-1}\mathbf{y} \end{aligned} \]

13.3.2.2 One to All

Consider the case in which the target population is partitioned into $N_I^B$ clusters, each of size $N_i^B$, $i=1,\ldots,N_I^B$. Each cluster of $U_I^B$ is associated with exactly one element $j$ of $U_A$. Note that $N_I^B=N^A$. Therefore, the link matrix is given by \[ \begin{aligned} \Theta_{AB}= \left( \begin{array}{cccc} \mathbf{1}'_{B1} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{1}'_{B2} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \ldots & \mathbf{1}'_{BN_I^B} \end{array} \right) \end{aligned} \]

where $\mathbf{1}_{Bi}$ is the vector of ones of size $N_i^B$, $i=1,\ldots,N_I$. The link matrix can also be written as $\Theta_{AB}=\textrm{diag}(\mathbf{1}'_{B1},\ldots,\mathbf{1}'_{BN_I^B}$; therefore, the standardized link matrix takes the following form: \[ \begin{aligned} \tilde{\Theta}_{AB}&=\Theta_{AB}[\textrm{diag}(\mathbf{1}_A\Theta_{AB})]^{-1}\\ &=\Theta_{AB}[\textrm{diag}(\mathbf{1}_A\textrm{diag}(\mathbf{1}'_{B1},\ldots,\mathbf{1}'_{BN_I^B}))]^{-1}\\ &=\Theta_{AB}[\textrm{diag}(\mathbf{1}'_{B1},\ldots,\mathbf{1}'_{BN_I^B})]^{-1}\\ &=\Theta_{AB}[\mathbf{I}_{\sum_{i=1}^{N_i}N_i^B\times \sum_{i=1}^{N_i}N_i^B}]^{-1}\\ &=\Theta_{AB} \end{aligned} \]

It follows that the vector of weights $\mathbf{w}$ is defined as \[ \begin{aligned} \mathbf{w}=\left(\dfrac{I_1^A}{\pi_1^A}\mathbf{1}_{B1},\ldots,\dfrac{I_{N_I^A}^A}{\pi_1^A}\mathbf{1}_{BN_I^B}\right)' \end{aligned} \]

and the estimator can be written as \[ \begin{aligned} \hat{t}_y&=\sum_{i=1}^{N_I^B}\dfrac{I_i^A}{\pi_i^A}t_{y,U_i} \end{aligned} \]

where $t_{y,U_i}=\sum_{k\in U_i^B}y_k$ is the total of the $i$-th cluster of population $U_I^B$.

13.3.2.3 All to One

In this case, population $U^A$ is considered partitioned into $N_I^A$ clusters, each of size $N_j^A$, $j=1,\ldots,N_I^A$. Each cluster of $U_I^A$ is associated with exactly one element $i$ of $U_B$. Note that $N_I^A=N^B$. Therefore, the link matrix is given by \[ \begin{aligned} \Theta_{AB}= \left( \begin{array}{cccc} \mathbf{1}_{A1} & \mathbf{0} & \cdots & \mathbf{0} \\ \mathbf{0} & \mathbf{1}_{A2} & \cdots & \mathbf{0} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0} & \mathbf{0} & \ldots & \mathbf{1}'_{AN_I^A} \end{array} \right) \end{aligned} \]

where $\mathbf{1}_{Aj}$ is the vector of ones of size $N_i^A$, $j=1,\ldots,N_I^A$. In this particular case, the standardized link matrix is given by the following expression: \[ \begin{aligned} \tilde{\Theta}_{AB}&=\Theta_{AB}[\textrm{diag}(\mathbf{1}_A\Theta_{AB})]^{-1}\\ &=\textrm{diag}(\dfrac{1}{N_1^A}\mathbf{1}_{A1},\ldots,\dfrac{1}{N_{N_I^A}^A}\mathbf{1}_{AN_I^A}) \end{aligned} \]

It follows that the vector of weights $\mathbf{w}$ is defined as \[ \begin{aligned} \mathbf{w}=\left(\dfrac{1}{N_1^A}\sum_{j\in U_I^A}\dfrac{I_j^A}{\pi_j^A},\ldots,\dfrac{1}{N_{N_I^A}^A}\sum_{j\in U_{N_I^A}^A}\dfrac{I_j^A}{\pi_j^A}\right)' \end{aligned} \]

and the resulting estimator takes the following form: \[ \begin{aligned} \hat{t}_y&=\sum_{i=1}^{N_I^A}\dfrac{y_i}{N_i^A}\sum_{j\in U_{i}^A}\dfrac{I_j^A}{\pi_j^A} \end{aligned} \]

13.4 Lexical-Graphic Example

Suppose that, in the example of the population of Parents and Children, whose link matrix is given by expression (14.1.2), a study is proposed to estimate the total for the Children. For this purpose, following the advice of Lavallé (2007), assume that the link matrix is given by \[ \Theta_{AB}= \left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \\ \end{array} \right) \]

In summary, the population of Parents $U_A$ is composed of $N_A=4$ individuals. The goal is to select a sample $S_A$ of $n_A=2$ individuals through a simple random sampling design. Under this configuration, there are $\binom{4}{2}=6$ possible samples. On the other hand, the target population $U_B$ is composed of $N_B=3$ Children whose ages are 2 years, 3 years, and 3 years, in that exact order. Thus, the population total of the characteristic of interest is $2+3+3=8$ years. Let us review each of the possible samples and see that, indeed, the resulting estimator is unbiased for the population total.

With this configuration, it is necessary to find the population standard link matrix. Thus, from Definition 14.2.1, we have \[ \theta_{+1}^{AB}=1, \ \ \ \ \theta_{+2}^{AB}=2, \ \ \ \ \theta_{+3}^{AB}=2 \]

Therefore, the standard link matrix is given by \[ \tilde{\Theta}_{AB}= \left( \begin{array}{ccc} 1 & 1/2 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/2 \\ 0 & 0 & 1/2 \\ \end{array} \right) \]

First sample: $s_A=\{Parent1, Parent2\}$. Parent1 links to Child1 and Child2, while Parent2 links only to Child2. In this way, the sample of the target population is defined as $s_B=\{Child1, Child2\}$. The weights are given below.

$w_1=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j1}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j1}=2(1+0)=2$
$w_2=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j2}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j2}=2(1/2+1/2)=2$
$w_3=0$ because no Parent linked to this child. After the observations are collected, the vector of values for the characteristic of interest is $y_s=(2,3)$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(2\times 2)+(2\times 3)=10$.

Second sample: $s_A=\{Parent1, Parent3\}$. Parent1 links to Child1 and Child2, while Parent3 links only to Child3. In this way, the sample of the target population is defined as $s_B=\{Child1, Child2, Child3\}$. The weights are given below.

$w_1=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j1}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j1}=2(1+0)=2$
$w_2=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j2}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j2}=2(1/2+0)=1$
$w_3=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j3}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j3}=2(0+1/2)=1$ After the observations are collected, the vector of values for the characteristic of interest is $y_s=(2,3,3)$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(2\times 2)+(1\times 3)+(1\times 3)=10$.

Third sample: $s_A=\{Parent1, Parent4\}$. Parent1 links to Child1 and Child2, while Parent4 links only to Child3. In this way, the sample of the target population is defined as $s_B=\{Child1, Child2, Child3\}$. The weights are given below.

$w_1=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j1}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j1}=2(1+0)=2$
$w_2=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j2}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j2}=2(1/2+0)=1$
$w_3=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j3}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j3}=2(0+1/2)=1$ After the observations are collected, the vector of values for the characteristic of interest is $y_s=(2,3,3)$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(2\times 2)+(1\times 3)+(1\times 3)=10$.

Fourth sample: $s_A=\{Parent2, Parent3\}$. Parent2 links only to Child2, while Parent4 links only to Child3. In this way, the sample of the target population is defined as $s_B=\{Child2, Child3\}$. The weights are given below.

$w_1=0$ because no Parent linked to this child.
$w_2=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j2}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j2}=2(1/2+0)=1$
$w_3=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j3}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j3}=2(0+1/2)=1$ After the observations are collected, the vector of values for the characteristic of interest is $y_s=(3,3)$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(1\times 3)+(1\times 3)=6$.

Fifth sample: $s_A=\{Parent2, Parent4\}$. Parent2 links only to Child2, while Parent4 links only to Child3. In this way, the sample of the target population is defined as $s_B=\{Child2, Child3\}$. The weights are given below.

$w_1=0$ because no Parent linked to this child.
$w_2=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j2}}{\pi_j^A}=(4/2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j2}=2(1/2+0)=1$
$w_3=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j3}}{\pi_j^A}=(2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j3}=2(0+1/2)=1$ After the observations are collected, the vector of values for the characteristic of interest is $y_s=(3,3)$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(1\times 3)+(1\times 3)=6$.

Sixth sample: $s_A=\{Parent3, Parent4\}$. Parent3 links only to Child3, as does Parent4. In this way, the sample of the target population is defined as $s_B=\{Child3\}$. The weights are given below.

$w_1=0$ because no Parent linked to this child.
$w_2=0$ because no Parent linked to this child.
$w_3=\sum_{j\in s_A}\frac{\tilde{\theta}^{AB}_{j3}}{\pi_j^A}=(2)\sum_{j\in s_A}\tilde{\theta}^{AB}_{j3}=2(1/2+1/2)=2$ After the observations are collected, the vector of values for the characteristic of interest is $y_s=3$, and the estimate is $\hat{t}_y=\sum_{i\in s_B}w_iy_i=(2\times 3)=6$.

In summary, averaging the estimates with respect to the sampling design $p_A$, it is easy to see that the estimator is unbiased because

\[ (1/6)\times(10+10+10+6+6+6)=8=t_y \]

On the other hand, note that the sampling design for the target population, which is unknown whenever only a single sample is selected, is given below:

\[ \begin{aligned} p_B(s_B)= \begin{cases} 2/6, & \text{if $s_B=\{Child1, Child2, Child3\}$}\\ 2/6, & \text{if $s_B=\{Child2, Child3\}$}\\ 1/6, & \text{if $s_B=\{Child1, Child2\}$}\\ 1/6, & \text{if $s_B=\{Child3\}$} \end{cases} \end{aligned} \]

Of course, in an illustrative exercise of this kind, it would be possible to calculate the inclusion probabilities and use the Horvitz-Thompson estimator to estimate the population total. However, in practice this option is quickly ruled out as the complexity of the sampling design and the sample size increase.

13.5 Exercises

Suppose that the total number of kilowatts consumed per month by households in a municipality needs to be estimated. Also assume that there is no household sampling frame, although there is a frame of individuals, and that to access the required information a sample of individuals is designed and they are asked about information for their household.

Argue why this problem can be solved with an indirect sampling approach.
Based on the above, propose an estimator for the total kilowatts consumed by households using the generalized weighting method.
If a sample of $n$ individuals was selected, define the inclusion probability of a household composed of $M<n$ individuals.
Based on the above, write the theoretical expressions for the Horvitz-Thompson and Hajek estimators for the total kilowatts consumed by households.

Under indirect sampling, propose an expression for the estimator of the population total if the sample $s_A$ was selected by simple random sampling.
Under indirect sampling, propose an expression for the estimator of the population total if the sample $s_A$ was selected by stratified random sampling.
Under indirect sampling, propose an expression for the estimator of the population total if the sample $s_A$ was selected using two-stage sampling.
Under indirect sampling, propose an expression for the estimator of the population total if a general regression estimator is used with auxiliary information characteristics from population $U_A$.
Under indirect sampling, propose an expression for the estimator of the population total if a general regression estimator is used with auxiliary information characteristics from population $U_B$.
What forms of nonresponse can occur in indirect sampling?
Discuss why network sampling can be viewed as a special case of indirect sampling and propose an estimator involving the generalized weighting method.
Discuss why adaptive sampling can be viewed as a special case of indirect sampling and propose an estimator involving the generalized weighting method.
Discuss why snowball sampling can be viewed as a special case of indirect sampling and propose an estimator involving the generalized weighting method.

Lavallé, P. 2007. Indirect Smapling. Springer.

Note that this is a special case of cluster sampling if the sampling frame of population $U_A$ were a cluster frame.↩︎
This restriction indicates that all members of population $U_A$ must have at least one link with some individual in the target population. Moreover, under this restriction, if there is some member of population $U_A$ that has no link with any member of population $U_B$, that member should not be considered.↩︎