  This paper is concerned with the formulation of $N$-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.
