Count data are omnipresent in many applied fields, often with overdispersion. With mixtures of Poisson distributions representing an elegant and appealing modelling strategy, we focus here on how the tail behaviour of the mixing distribution is related to the tail of the resulting Poisson mixture. We define five sets of mixing distributions and we identify for each case whenever the Poisson mixture is in, close to or far from a domain of attraction of maxima. We also characterize how the Poisson mixture behaves similarly to a standard Poisson distribution when the mixing distribution has a finite support. Finally, we study, both analytically and numerically, how goodness-of-fit can be assessed with the inspection of tail behaviour.

翻译：计数数据在许多应用领域无处不在，且常伴随过离散现象。鉴于泊松分布的混合模型是一种优雅且富有吸引力的建模策略，本文重点研究混合分布的尾部行为如何影响其生成的泊松混合分布的尾部特性。我们定义了五类混合分布，并针对每种情况确定了泊松混合分布何时处于最大值吸引域内、接近该域或远离该域。同时，我们刻画了当混合分布具有有限支撑时，泊松混合分布如何类似于标准泊松分布。最后，我们通过解析与数值方法，研究了如何通过检验尾部行为来评估拟合优度。