Count time series data are frequently analyzed by modeling their conditional means and the conditional variance is often considered to be a deterministic function of the corresponding conditional mean and is not typically modeled independently. We propose a semiparametric mean and variance joint model, called random rounded count-valued generalized autoregressive conditional heteroskedastic (RRC-GARCH) model, to address this limitation. The RRC-GARCH model and its variations allow for the joint modeling of both the conditional mean and variance and offer a flexible framework for capturing various mean-variance structures (MVSs). One main feature of this model is its ability to accommodate negative values for regression coefficients and autocorrelation functions. The autocorrelation structure of the RRC-GARCH model using the proposed Laplace link functions with nonnegative regression coefficients is the same as that of an autoregressive moving-average (ARMA) process. For the new model, the stationarity and ergodicity are established and the consistency and asymptotic normality of the conditional least squares estimator are proved. Model selection criteria are proposed to evaluate the RRC-GARCH models. The performance of the RRC-GARCH model is assessed through analyses of both simulated and real data sets. The results indicate that the model can effectively capture the MVS of count time series data and generate accurate forecast means and variances.

翻译：计数时间序列数据通常通过建模条件均值进行分析，而条件方差常被视为对应条件均值的确定性函数，一般不单独建模。针对这一局限性，我们提出了一种半参数均值与方差联合模型——随机取整计数广义自回归条件异方差（RRC-GARCH）模型。RRC-GARCH模型及其变体允许同时建模条件均值与方差，为捕捉多种均值-方差结构（MVS）提供了灵活框架。该模型的主要特征之一是其能够适应回归系数与自相关函数的负值。采用非负回归系数的拉普拉斯链接函数所构建的RRC-GARCH模型，其自相关结构与自回归滑动平均（ARMA）过程相同。针对该新模型，我们建立了平稳性与遍历性，并证明了条件最小二乘估计量的一致性与渐近正态性。提出了模型选择准则以评估RRC-GARCH模型。通过模拟与真实数据集的分析，评估了RRC-GARCH模型的性能。结果表明，该模型能有效捕捉计数时间序列数据的MVS，并生成准确的均值与方差预测。