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Summary: This paper is concerned with the formulation of N$$ N $$‐mixture models for estimating the abundance and probability of detection of a species from binary response, count and time‐to‐detection data. A modelling framework, which encompasses time‐to‐first‐detection within the context of detection/non‐detection and time‐to‐each‐detection and time‐to‐first‐detection within the context of count data, is introduced. Two observation processes which depend on whether or not double counting is assumed to occur are also considered. The main focus of the paper is on the derivation of explicit forms for the likelihoods associated with each of the proposed models. Closed‐form expressions for the likelihoods associated with time‐to‐detection data are new and are developed from the theory of order statistics. A key finding of the study is that, based on the assumption of no double counting, the likelihoods associated with times‐to‐detection together with count data are the product of the likelihood for the counts alone and a term which depends on the detection probability parameter. This result demonstrates that, in this case, recording times‐to‐detection could well improve precision in estimation over recording counts alone. In contrast, for the double counting protocol with exponential arrival times, no information was found to be gained by recording times‐to‐detection in addition to the count data. An R package and an accompanying vignette are also introduced in order to complement the algebraic results and to demonstrate the use of the models in practice. [ABSTRACT FROM AUTHOR]

Multinomial N‐mixture models are commonly used to fit data from a removal sampling protocol. If the mixing distribution is negative binomial, the distribution of the counts does not appear to have been identified, and practitioners approximate the requisite likelihood by placing an upper bound on the embedded infinite sum. In this paper, the distribution which underpins the multinomial N‐mixture model with a negative binomial mixing distribution is shown to belong to the broad class of multivariate negative binomial distributions. Specifically, the likelihood can be expressed in closed form as the product of conditional and marginal likelihoods and the information matrix shown to be block diagonal. As a consequence, the nature of the maximum likelihood estimates of the unknown parameters and their attendant standard errors can be examined and tests of the hypothesis of the Poisson against the negative binomial mixing distribution formulated. In addition, appropriate multinomial N‐mixture models for data sets which include zero site totals can also be constructed. Two illustrative examples are provided. [ABSTRACT FROM AUTHOR]

The focus of this article is on the nature of the likelihood associated with N-mixture models for repeated count data. It is shown that the infinite sum embedded in the likelihood associated with the Poisson mixing distribution can be expressed in terms of a hypergeometric function and, thence, in closed form. The resultant expression for the likelihood can be readily computed to a high degree of accuracy and is algebraically tractable. Specifically, the likelihood equations can be simplified to some advantage, the concentrated likelihood in the probability of detection formulated and problematic cases identified. The results are illustrated by means of a simulation study and a real world example. The study is extended to N-mixture models with a negative binomial mixing distribution and results similar to those for the Poisson case obtained. N-mixture models with mixing distributions which accommodate excess zeros and, separately, with a beta-binomial distribution rather than a binomial used to model the intra-site counts are also investigated. However the results for these settings, while computationally attractive, do not provide insight into the nature of the maximum likelihood estimates. [ABSTRACT FROM AUTHOR]