Max-autogressive moving average (Max-ARMA) processes are powerful tools for modelling time series data with heavy-tailed behaviour; these are a non-linear version of the popular autoregressive moving average models. River flow data typically have features of heavy tails and non-linearity, as large precipitation events cause sudden spikes in the data that then exponentially decay. Therefore, stationary Max-ARMA models are a suitable candidate for capturing the unique temporal dependence structure exhibited by river flows. This paper contributes to advancing our understanding of the extremal properties of stationary Max-ARMA processes. We detail the first approach for deriving the extremal index, the lagged asymptotic dependence coefficient, and an efficient simulation for a general Max-ARMA process. We use the extremal properties, coupled with the belief that Max-ARMA processes provide only an approximation to extreme river flow, to fit such a model which can broadly capture river flow behaviour over a high threshold. We make our inference under a reparametrisation which gives a simpler parameter space that excludes cases where any parameter is non-identifiable. We illustrate results for river flow data from the UK River Thames.

翻译：最大自回归移动平均（Max-ARMA）过程是建模具有重尾行为时间序列数据的强大工具；它是流行的自回归移动平均模型的非线性版本。河流流量数据通常具有重尾和非线性的特征，因为大的降水事件会导致数据中出现突然的峰值，随后呈指数衰减。因此，平稳Max-ARMA模型是捕捉河流流量独特时间依赖结构的合适候选。本文有助于增进我们对平稳Max-ARMA过程极值特性的理解。我们详细介绍了推导一般Max-ARMA过程极值指数、滞后渐近依赖系数以及高效模拟的首种方法。利用极值特性，并结合Max-ARMA过程仅提供极端河流流量近似的观点，我们拟合了这样一个模型，该模型能够大致捕捉高阈值之上的河流流量行为。我们在一种重新参数化下进行推断，该参数化给出了一个更简单的参数空间，排除了任何参数不可识别的情况。我们展示了英国泰晤士河流量数据的结果。