Inference on population size in binomial detectability models

Abstract

Many models for biological populations, including simple mark-recapture models and distance sampling models, involve a binomially distributed number, n, of observations x1, …, xn on members of a population of size N. Two popular estimators of (N, θ), where θ is a vector parameter, are the maximum likelihood estimator and the conditional maximum likelihood estimator based on the conditional distribution of x1, …, xn given n. We derive the large-N asymptotic distributions of and ⁠, and give formulae for the biases of and ⁠. We show that the difference is, remarkably, of order 1 and we give a simple formula for the leading part of this difference. Simulations indicate that in many cases this formula is very accurate and that confidence intervals based on the asymptotic distribution have excellent coverage. An extension to product-binomial models is given.