50-Year-Old Curiosities: Ancillarity and Inference in Capture–Recapture Models

Abstract

We review developments from the late 1950s, starting with the work of John Darroch, that led to the models of Cormack [Biometrika 51 (1964) 429–438], Jolly [Biometrika 52 (1965) 225–247] and Seber [Biometrika 52 (1965) 249–259] that are commemorated in this volume. We emphasize some of the fundamental contributions that were pivotal and often ahead of their time. We look at how these early contributions helped to shape the field and illustrate important concepts in statistics, including sufficiency, ancillarity, partial likelihoods, missing data and model fitting in the presence of latent variables. We also identify two curiosities. The first is the long-held and mistaken belief that the maximum likelihood estimators for various capture–recapture models are in common. Using various notions of ancillarity, we show that the maximum likelihood estimators from partial models (like the Cormack–Jolly–Seber model) will in general differ from full likelihood approaches (such as the Jolly–Seber model). The second is the belief that model specification in terms of latent variables is a relatively recent advance. We highlight how Jolly (1965) used a state-space model to describe the problem, using latent variables to separate the capture and mortality processes. We show how Markov chain Monte Carlo can be used to fit this model and how it relates to other capture–recapture models specified in terms of latent variables.