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F\'elix-Medina and Thompson (2004) proposed a variant of link-tracingsampling to estimate the size of a hidden population such as drug users, sexualworkers or homeless people. In their variant a sampling frame of sites wherethe members of the population tend to gather is constructed. The frame is notassumed to cover the whole population, but only a portion of it. A simplerandom sample of sites is selected; the people in the sampled sites areidentified and are asked to name other members of the population which areadded to the sample. Those authors proposed maximum likelihood estimators ofthe population size which derived from a multinomial model for the numbers ofpeople found in the sampled sites and a model that considers that theprobability that a person is named by any element in a particular sampled site(link-probability) does not depend on the named person, that is, that theprobabilities are homogeneous. Later, F\'elix-Medina et al. (2015) proposedunconditional and conditional maximum likelihood estimators of the populationsize which derived from a model that takes into account the heterogeneity ofthe link-probabilities. In this work we consider this sampling design and setconditions for a general model for the link-probabilities that guarantee theconsistency and asymptotic normality of the estimators of the population sizeand of the estimators of the parameters of the model for thelink-probabilities. In particular we showed that both the unconditional andconditional maximum likelihood estimators of the population size are consistentand have asymptotic normal distributions which are different from each other.

Combining cluster sampling and link-tracing sampling to estimate the size of a hidden population: asymptotic properties of the estimators