In the analysis of real integer-valued time series data, we often encounter negative values and negative correlations. For integer-valued autoregressive time series, there are many parametric models to choose from, but some of them are relatively complex. With little information about the background of real data, we hope that a simple and effective semiparametric model can be used to obtain more information that usually cannot be provided by parametric models, such as the confidence interval of the innovation distribution. But the only existing semiparametric model based on thinning operators can only deal with non-negative data with positive correlation coefficients. In addition, it has two drawbacks: first, an initial distribution of the innovation is required, but different initial values may lead to different results; second, the confidence interval of the innovation distribution is not available, which is essential in low-valued data. To overcome these drawbacks, we propose a rounded semiparametric autoregressive model with a log-concave innovation, which can deal with $\mathbb{Z}$-valued time series with autoregressive coefficients of arbitrary sign. The consistencies of the estimators for the parametric and nonparametric parts of the model are also discussed. We illustrate the superior performance of the proposed model based on three real datasets.

中文翻译：

---

ℤ 上的半参数整数值自回归模型

在分析实整数值时间序列数据时，我们经常会遇到负值和负相关。对于整数值自回归时间序列，有许多参数模型可供选择，但其中一些相对复杂。在关于真实数据背景的信息很少的情况下，我们希望可以使用简单有效的半参数模型来获得更多通常参数模型无法提供的信息，例如创新分布的置信区间。但是现有的唯一基于细化算子的半参数模型只能处理具有正相关系数的非负数据。此外，它还有两个缺点：第一，需要对创新进行初始分布，但不同的初始值可能会导致不同的结果；第二，创新分布的置信区间不可用，这在低值数据中是必不可少的。为了克服这些缺点，我们提出了一个带有对数凹创新的圆角半参数自回归模型，它可以处理$\mathbb{Z}$具有任意符号自回归系数的值时间序列。还讨论了模型的参数和非参数部分的估计量的一致性。我们基于三个真实数据集说明了所提出模型的卓越性能。