An empirical model for underdispersed count data

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Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts (2016, methodology): “Other alternatives, such as the log-linear hyperPoisson regression model (Sáez-Castillo & Conde-Sánchez, 2013), which is a special case of the exponentially-weighted Poisson model (Ridout & Besbeas, 2004), and the Gamma-count model (Zeviani et al., 2014), have also been proposed recently.”
Integer-valued autoregressive models for counts showing underdispersion (2013, methodology): “Also χ2-values are shown (computed by using the respective ML-fitted model) such that a comparison with [3,25] is possible.”
Integrating demographic data: towards a framework for monitoring wildlife populations at large spatial scales (2014, methodology): “We modelled brood size using an exponentially weighted Poisson distribution (EWP), which provides an empirical fit to under-dispersed discrete data (Ridout & Besbeas 2004; Robinson, Baillie & King 2012).”