  Respondent-driven sampling is a survey method for hidden or hard-to-reach
populations in which sampled individuals recruit others in the study population
via their social links. The most popular estimator for for the population mean
assumes that individual sampling probabilities are proportional to each
subject's reported degree in a social network connecting members of the hidden
population. However, it remains unclear under what circumstances these
estimators are valid, and what assumptions are formally required to identify
population quantities. In this short note we detail nonparametric
identification results for the population mean when the sampling probability is
assumed to be a function of network degree known to scale. Importantly, we
establish general conditions for the consistency of the popular Volz-Heckathorn
(VH) estimator. Our results imply that the conditions for consistency of the VH
estimator are far less stringent than those suggested by recent work on
diagnostics for RDS. In particular, our results do not require random sampling
or the existence of a network connecting the population.
