The analysis of high-dimensional time series data has become increasingly important across a wide range of fields. Recently, a method for constructing the minimum information Markov kernel on finite state spaces was established. In this study, we propose a statistical model based on a parametrization of its dependence function, which we call the \textit{Minimum Information Markov Model}. We show that its parametrization induces an orthogonal structure between the stationary distribution and the dependence function, and that the model arises as the optimal solution to a divergence rate minimization problem. In particular, for the Gaussian autoregressive case, we establish the existence of the optimal solution to this minimization problem, a nontrivial result requiring a rigorous proof. For parameter estimation, our approach exploits the conditional independence structure inherent in the model, which is supported by the orthogonality. Specifically, we develop several estimators, including conditional likelihood and pseudo likelihood estimators, for the minimum information Markov model in both univariate and multivariate settings. We demonstrate their practical performance through simulation studies and applications to real-world time series data.

翻译：高维时间序列数据的分析在众多领域中正变得日益重要。最近，一种在有限状态空间上构建最小信息马尔可夫核的方法被提出。在本研究中，我们提出了一种基于其依赖函数参数化的统计模型，我们称之为\textit{最小信息马尔可夫模型}。我们证明了其参数化在平稳分布与依赖函数之间诱导出一种正交结构，并且该模型是散度率最小化问题的最优解。特别地，对于高斯自回归情形，我们建立了该最小化问题最优解的存在性，这是一个需要严格证明的非平凡结果。在参数估计方面，我们的方法利用了模型固有的条件独立结构，该结构由正交性所支撑。具体而言，我们为最小信息马尔可夫模型在单变量和多变量设定下开发了多种估计量，包括条件似然估计量和伪似然估计量。我们通过模拟研究和真实世界时间序列数据的应用，展示了这些估计量的实际性能。