In the past four decades, research on count time series has made significant progress, but research on $\mathbb{Z}$-valued time series is relatively rare. Existing $\mathbb{Z}$-valued models are mainly of autoregressive structure, where the use of the rounding operator is very natural. Because of the discontinuity of the rounding operator, the formulation of the corresponding model identifiability conditions and the computation of parameter estimators need special attention. It is also difficult to derive closed-form formulae for crucial stochastic properties. We rediscover a stochastic rounding operator, referred to as mean-preserving rounding, which overcomes the above drawbacks. Then, a novel class of $\mathbb{Z}$-valued ARMA models based on the new operator is proposed, and the existence of stationary solutions of the models is established. Stochastic properties including closed-form formulae for (conditional) moments, autocorrelation function, and conditional distributions are obtained. The advantages of our novel model class compared to existing ones are demonstrated. In particular, our model construction avoids identifiability issues such that maximum likelihood estimation is possible. A simulation study is provided, and the appealing performance of the new models is shown by several real-world data sets.

翻译：过去四十年中，计数时间序列的研究取得了显著进展，但关于$\mathbb{Z}$值时间序列的研究相对较少。现有$\mathbb{Z}$值模型主要基于自回归结构，其中取整算子的应用非常自然。由于取整算子的非连续性，相应模型可识别性条件的制定以及参数估计量的计算需要特别关注，同时关键随机性质的闭式公式推导也面临困难。我们重新发现了一种随机取整算子（称为均值保持取整），它克服了上述缺陷。随后，基于该新算子提出了一类新颖的$\mathbb{Z}$值ARMA模型，并建立了模型平稳解的存在性。我们获得了包括（条件）矩、自相关函数和条件分布在内的闭式公式在内的随机性质，并证明了新模型相对于现有模型的优势。特别地，我们的模型构造避免了可识别性问题，使得极大似然估计成为可能。通过模拟研究验证了模型性能，并基于多个真实数据集展示了新模型的优越表现。