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N-mixture models are probability models that estimate abundance using replicate observed counts while accounting for imperfect detection. In this article, we propose an asymptotic approximation to the N-mixture model which efficiently estimates large abundances without the computational limitations of the generalized N-mixture model introduced by Dail and Madsen in 2011. It has been suggested in the literature that N-mixture models do not perform well when counts from the same sites show weak patterns of population dynamics. Our proposed model addresses this issue by using the asymptotic information matrix to diagnose model parameter estimability and to derive parameter standard errors. A simulation study show that this model performs almost as well as the Dail-Madsen Generalized N-mixture model at low abundances and improves on it at higher abundances. We illustrate the procedure using two data sets: the American robin data from Dail and Madsen (2011), and counts of chlamydia cases in the state of Oregon from 2007-2016. The chlamydia data exhibit very large abundances and demonstrate the potential usefulness of the proposed model for disease surveillance data., (© 2018, The International Biometric Society.)