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We propose a modern method to estimate population size based oncapture-recapture designs of K samples. The observed data is formulated as asample of n i.i.d. K-dimensional vectors of binary indicators, where the k-thcomponent of each vector indicates the subject being caught by the k-th sample,such that only subjects with nonzero capture vectors are observed. The targetquantity is the unconditional probability of the vector being nonzero acrossboth observed and unobserved subjects. We cover models assuming a singleconstraint (identification assumption) on the K-dimensional distribution suchthat the target quantity is identified and the statistical model isunrestricted. We present solutions for linear and non-linear constraintscommonly assumed to identify capture-recapture models, including no K-wayinteraction in linear and log-linear models, independence or conditionalindependence. We demonstrate that the choice of constraint has a dramaticimpact on the value of the estimand, showing that it is crucial that theconstraint is known to hold by design. For the commonly assumed constraint ofno K-way interaction in a log-linear model, the statistical target parameter isonly defined when each of the $2^K - 1$ observable capture patterns is present,and therefore suffers from the curse of dimensionality. We propose a targetedMLE based on undersmoothed lasso model to smooth across the cells whiletargeting the fit towards the single valued target parameter of interest. Foreach identification assumption, we provide simulated inference and confidenceintervals to assess the performance on the estimator under correct andincorrect identifying assumptions. We apply the proposed method, alongsideexisting estimators, to estimate prevalence of a parasitic infection usingmulti-source surveillance data from a region in southwestern China, under thefour identification assumptions.

Estimation of population size based on capture recapture designs and evaluation of the estimation reliability