The paper introduces a general class of heavy‐tailed auto‐regressions for modelling integer‐valued time series with outliers. The specification proposed is based on a heavy‐tailed mixture of negative binomial distributions that features an observation‐driven dynamic equation for the conditional expectation. The existence of a stationary and ergodic solution for the class of auto‐regressive processes is shown under general conditions. The estimation of the model can be easily performed by maximum likelihood given the closed form of the likelihood function. The strong consistency and the asymptotic normality of the estimator are formally derived. Two examples of specifications illustrate the flexibility of the approach and the relevance of the theoretical results. In particular, a linear dynamic equation and a score‐driven equation for the conditional expectation are studied. The score‐driven specification is shown to be particularly appealing as it delivers a robust filtering method that attenuates the effect of outliers. Empirical applications to the series of narcotics trafficking reports in Sydney and the euro–pound sterling exchange rate illustrate the effectiveness of the method in handling extreme observations.

中文翻译：

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Beta负二项式自回归，用于使用极端观测值对整数值时间序列建模

本文介绍了一种通用类的重尾自回归，用于建模具有离群值的整数值时间序列。提出的规范基于负二项式分布的重尾混合，其中具有条件期望的观察驱动动态方程。在一般条件下，证明了一类自回归过程的平稳解和遍历解的存在。给定似然函数的封闭形式，可以通过最大似然来轻松执行模型的估计。正式推导了估计的强一致性和渐近正态性。规格的两个例子说明了该方法的灵活性和理论结果的相关性。尤其是，研究了用于条件期望的线性动力学方程和分数驱动方程。分数驱动的规范特别吸引人，因为它提供了一种强大的过滤方法，可以减弱异常值的影响。在悉尼的一系列麻醉品贩运报告以及欧元兑英镑汇率的经验应用中，证明了该方法在处理极端观察结果方面的有效性。