Inference on population size in binomial detectability models
Many models for biological populations, including simple mark-recapture models and distance sampling models, involve a binomially distributed number, n, of observations x<sub>1</sub>, …, x<sub>n</sub> 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 <inline-formula><inline-graphic xlink:href="asp051ilm1.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> and the conditional maximum likelihood estimator <inline-formula><inline-graphic xlink:href="asp051ilm2.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> based on the conditional distribution of x<sub>1</sub>, …, x<sub>n</sub> given n. We derive the large-N asymptotic distributions of <inline-formula><inline-graphic xlink:href="asp051ilm3.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>and <inline-formula><inline-graphic xlink:href="asp051ilm4.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>, and give formulae for the biases of <inline-formula><inline-graphic xlink:href="asp051ilm5.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> and <inline-formula><inline-graphic xlink:href="asp051ilm6.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>. We show that the difference <inline-formula><inline-graphic xlink:href="asp051ilm7.gif" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>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. Copyright 2009, Oxford University Press.
