We present a novel approach for modeling bounded count time series data, by deriving accurate upper and lower bounds for the variance of a bounded count random variable while maintaining a fixed mean. Leveraging these bounds, we propose semiparametric mean and variance joint (MVJ) models utilizing a clipped-Laplace link function. These models offer a flexible and feasible structure for both mean and variance, accommodating various scenarios of under-dispersion, equi-dispersion, or over-dispersion in bounded time series. The proposed MVJ models feature a linear mean structure with positive regression coefficients summing to one and allow for negative regression cefficients and autocorrelations. We demonstrate that the autocorrelation structure of MVJ models mirrors that of an autoregressive moving-average (ARMA) process, provided the proposed clipped-Laplace link functions with nonnegative regression coefficients summing to one are utilized. We establish conditions ensuring the stationarity and ergodicity properties of the MVJ process, along with demonstrating the consistency and asymptotic normality of the conditional least squares estimators. To aid model selection and diagnostics, we introduce two model selection criteria and apply two model diagnostics statistics. Finally, we conduct simulations and real data analyses to investigate the finite-sample properties of the proposed MVJ models, providing insights into their efficacy and applicability in practical scenarios.

翻译：我们提出了一种对有界计数时间序列数据进行建模的新方法，通过推导有界计数随机变量在固定均值条件下的方差精确上下界，基于这些边界，我们利用剪切拉普拉斯连接函数构建了半参数均值方差联合模型。该模型为均值和方差提供了灵活可行的结构，能够适应有界时间序列中的欠分散、等分散或过分散等各种情形。所提出的均值方差联合模型采用线性均值结构，其回归系数为非负且和为1，同时允许负回归系数与负自相关存在。我们证明，当采用所提出的非负回归系数之和为1的剪切拉普拉斯连接函数时，该模型的自相关结构与自回归滑动平均过程相似。我们建立了确保均值方差联合过程平稳性与遍历性的条件，并证明条件最小二乘估计量的相合性与渐近正态性。为辅助模型选择与诊断，我们引入两种模型选择准则并应用两种模型诊断统计量。最后通过模拟与真实数据分析，考察所提出模型在有限样本下的性质，进而揭示其在实际场景中的有效性与适用性。