Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities 2. Sampling design and notation
Since in this work we consider the variant of LTS proposed by Félix-Medina and Thompson (2004), we will briefly describe it. Thus, let be a finite population of an unknown number of people. We assume that a portion of is covered by a sampling frame of sites where the members of the population can be found with high probability. We suppose that we have a criterion that allows us to assign a person in to only one site in the frame. Notice that we are not assuming that a person could not be found in different sites, but that, as in ordinary cluster sampling, we are able to assign him or her to only one site, for instance, the site where he or she spends most of his or her time. Thus, we can consider the sites in the frame as clusters of people. Let denote the number of members of the population that belong to the site From the previous assumption it follows that the number of people in is and the number of people in the portion of that is not covered by the frame is
The sampling design is as follows. A SRSWOR of sites is selected from the frame and the members of the population who belong to the sampled site are identified, Let be the set of people in the initial sample. Observe that the size of is The people in each sampled site are asked to name other members of the population. We will say that a person and a site are linked if any of the people who belong to that site names him or her. Finally, let and be the sets of people in and respectively, who are linked to some site or sites in
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