A popular and flexible time series model for counts is the generalized integer autoregressive process of order $p$, GINAR($p$). These Markov processes are defined using thinning operators evaluated on past values of the process along with a discretely-valued innovation process. This class includes the commonly used INAR($p$) process, defined with binomial thinning and Poisson innovations. GINAR processes can be used in a variety of settings, including modeling time series with low counts, and allow for more general mean-variance relationships, capturing both over- or under-dispersion. While there are many thinning operators and innovation processes given in the literature, less focus has been spent on comparing statistical inference and forecasting procedures over different choices of GINAR process. We provide an extensive study of exact and approximate inference and forecasting methods that can be applied to a wide class of GINAR($p$) processes with general thinning and innovation parameters. We discuss the challenges of exact estimation when $p$ is larger. We summarize and extend asymptotic results for estimators of process parameters, and present simulations to compare small sample performance, highlighting how different methods compare. We illustrate this methodology by fitting GINAR processes to a disease surveillance series.

翻译：一种流行且灵活的计数时间序列模型是阶数为$p$的广义整数自回归过程，即GINAR($p$)。这些马尔可夫过程通过将稀疏算子应用于过程的过去值以及离散取值的创新过程来定义。该类别包括常用的INAR($p$)过程，该过程通过二项稀疏和泊松创新定义。GINAR过程可应用于多种场景，包括低计数时间序列建模，并允许更一般的均值-方差关系，同时捕捉过度分散或欠分散现象。尽管文献中提出了许多稀疏算子和创新过程，但针对不同GINAR过程选择的统计推断与预测方法的比较研究较少。我们针对可应用于具有一般稀疏与创新参数的大类GINAR($p$)过程，开展了精确与近似推断及预测方法的全面研究。讨论了当$p$较大时精确估计的挑战。总结并扩展了过程参数估计量的渐进结果，通过仿真比较小样本性能，突出不同方法的优劣。最后通过将GINAR过程拟合至疾病监测序列，展示了该方法的实际应用。