Doubly robust capture-recapture methods for estimating population size

Estimation of population size using incomplete lists has a long history across many biological and social sciences. For example, human rights groups often construct partial lists of victims of armed conflicts, to estimate the total number of victims. Earlier statistical methods for this setup often use parametric assumptions, or rely on suboptimal plug-in-type nonparametric estimators; but both approaches can lead to substantial bias, the former via model misspecification and the latter via smoothing. Under an identifying assumption that two lists are conditionally independent given measured covariates, we make several contributions. First, we derive the nonparametric efficiency bound for estimating the capture probability, which indicates the best possible performance of any estimator, and sheds light on the statistical limits of capture-recapture methods. Then we present a new estimator, that has a double robustness property new to capture-recapture, and is near-optimal in a non-asymptotic sense, under relatively mild nonparametric conditions. Next, we give a confidence interval construction method for total population size from generic capture probability estimators, and prove non-asymptotic near-validity. Finally, we apply them to estimate the number of killings and disappearances in Peru during its internal armed conflict between 1980 and 2000.