and the incorporation of covariates. They develop Bayesian inference for their specification. Manrique-Vallier and Fienberg (2008) expand on this approach, modeling individual-level heterogeneity using a Grade of Membership model wherein individuals are postulated as mixtures of latent homogeneous but extreme “ideal” types.
Many populations, including that of unauthorized crossers, are open in the sense that the population experiences change during or between the sampling (e.g., births, deaths). Many of the models reviewed above implicitly presume the population is closed (i.e., have fixed and unchanging membership). For open populations, interest typically has focused on the case where the population is closed during the period of each capture and experiences immigration and mortality between the capture periods. Cormack (1989) reviews many of the classical models for this case. Pledger and colleagues (2003) extend these to allow for individual heterogeneity in survival and capture rates using a finite mixture formulation. These models are receiving continuous development (see the review by Royle and Dorazio ).
CAPTURE-RECAPTURE APPLICATIONS TO UNAUTHORIZED BORDER CROSSINGS
The most direct expression of capture-recapture ideas as applied to unauthorized border crossings is the work of Espenshade (1990, 1995b) and Singer and Massey (1998). They develop simple CRC models in the context of apprehensions (“capture”) and re-apprehension (“recapture”) of unauthorized crossers. Specifically, Espenshade (1995b) models as a geometric distribution the number of crossings an individual makes until a successful crossing. Under assumptions that individuals continue to attempt crossings until they succeed, that the probability of success is the same for each attempt, and other strong assumptions, he derives the equivalent of the Petersen estimator for the number of unauthorized crossers. He does not develop measures of uncertainty of this estimate, nor does he tie the work into the broader CRC literature. This approach is similar in spirit to that of the “frequency of apprehension frequencies” discussed in Chapter 5. Chang and colleagues (2006) extend these methods to treat “discouragement” due to prior apprehension and “return and rentry” due to unobserved exit and reentry into the United States. However, the panel did not have access to their paper and therefore could not review it; the only available description was by Morral and colleagues (2011).
A variant of CRC is “red teaming,” in which individuals are recruited to attempt to cross so as to get an estimate of the probability of apprehension. This is referred to as plant-capture in the ecological literature (Goudie et al., 2007).