Alessio Farcomeni (University of Rome La Sapienza)

Semiparametric capture-recapture with heterogeneous capture probabilities

Capture-recapture experiments are commonly used to estimate the size of a closed population. Link (2003) has underlined identifiability problems when one wants to make inference with heterogeneous capture probabilities in a semiparametric framework. If subject-specific capture probabilities are random effects with no assumption on the mixing distribution, the conditional likelihood is not identifiable. Link (2003) invokes the equivalence of conditional and complete likelihood (Sanathanan, 1972) to conclude that semiparametric inference is not possible in recapture studies. We show (i) that a regularity condition of Sanathanan (1972) is not met, and hence such equivalence does not hold; (ii) that the complete likelihood is indeed identifiable. Surprisingly enough, we also show that the MLE is convergent but not consistent. We characterize the limiting value of the MLE as a “sharpest estimable lower bound” and prove that the MLE can never over estimate the true population size. In practice, the complete likelihood parameter space includes all possible mixing distributions with support on [0,1]. We use the theory of canonical moments and a logistic transform to obtain a finite dimensional and unconstrained parameter space. We then underline computational and philosophical problems related to maximum likelihood estimation. Consequently, we propose a Bayesian approach which is extended to the most general model in recapture studies, where heterogeneous detection probabilities are allowed also to depend on trapping occasion and behavioural reactions to first capture. We derive the Jeffrey’s prior and illustrate the Bayesian and classical approaches with real examples and simulations.

Joint work with Luca Tardella.