A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the Expectation-Maximization (EM) algorithm are proposed for estimating parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed model. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-taildness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. An application to modeling bivariate count time series data on cannabis possession-related offenses in Australia is discussed.

翻译：提出了一种基于多变量泊松广义逆高斯分布的新型多变量整数值广义自回归条件异方差过程。由于似然函数包含依赖于多变量计数及其维度的贝塞尔函数，通过极大似然方法估计所提出的多变量重尾计数时间序列模型参数具有挑战性，这导致优化过程中常出现数值不稳定性。为克服该计算难题，针对低维和高维设定，提出了两种可行的期望最大化算法变体来估计模型参数。这些EM算法变体具有计算优势，避免了直接优化所提模型似然函数的困难。本模型及其估计方法能处理多变量计数建模、重尾性、过度离散、异常值包容、正负自相关、交叉/同期相关估计，以及从统计与计算双重角度的参数高效估计等多项特性。通过广泛的蒙特卡洛仿真研究评估了所提EM算法的性能，并讨论了其在澳大利亚大麻持有相关犯罪双变量计数时间序列数据建模中的应用。