高维整数值时间序列的建模、统计推断及应用研究

High-dimensional integer-valued time series data is one of the common data types in the age of big data, its modeling is a difficult task in analyzing data, but there is a growing interest in it. Existing models for modeling one-dimensional data are integer-valued autoregression and integer-valued GARCH, and their generalization to low-dimensional cases (the dimension is less than 5) are based on multivariate discrete distributions or copula functions, but research on high-dimensional cases is blank, modeling methods for low-dimensional models are generalized directly to high-dimensional cases will result in lack of identification, noisy estimates, unstable predictions and difficult-to-interpret temporal dependence, while sparsity assumption on matrix of coefficient and introducing factor structure can avoid above shortcoming. This project studies high-dimensional integer-valued autoregression and high-dimensional integer-valued GARCH models, and gives several modeling strategies for each kind of model. It gives stationarity and ergodicity, conditions for identification and criteria for determining the number of factors, and discusses estimating parameters and establishes large-sample properties for estimators (including consistency, rate of convergence and asymptotic normality), and considers numerical algorithm and hypothesis test in high-dimensional setting, and studies high-dimensional integer-valued data sets in finance and insurance using the proposed approaches. These studies are not only very important both in theory and application, but also fill in a gap in research on high-dimensional integer-valued time series.

高维整数值时间序列数据是大数据时代常见的数据类型之一，它的建模是数据分析中的一个难点，但相关的兴趣一直在增加。建模一维数据的已有模型是整数值自回归和整数值GARCH，其低维(维数不超过5)推广是基于多元离散分布或者copula函数，但高维模型的研究还是空白，将低维的建模方法直接推广到高维会导致缺少识别性、有噪声的参数估计、不稳定的预测和难于解释的时间相依性，系数矩阵的稀疏化假设和引入因子结构可以解决上述缺点。本项目研究高维整数值自回归和高维整数值GARCH模型，针对每类模型给出多种建模策略。给出新模型的平稳性和遍历性、识别条件和确定因子个数的准则，讨论参数估计并建立估计量的大样本性质(包括相合性、收敛速率和渐近正态性)，考虑高维情形的数值算法、假设检验，利用提出的方法研究金融和保险中的高维整数值数据。这些研究成果不仅具有重要的理论和应用价值，还将填补高维整数值时间序列研究的空白。

高维整数值时间序列数据是大数据时代常见的数据类型之一，它的建模是数据分析中的一个难点，但相关的兴趣一直在增加。本项目研究了高维整数值自回归和高维整数值GARCH模型的多种建模策略。已取得的成果包括：计数值INAR和INGARCH模型的建模方法和推断、Z值INAR和INGARCH模型的建模方法和推断、白噪声的检验、结构变点和门限的检验，已经发表相关SCI论文36篇。我们给出了新模型的平稳性和遍历性，讨论了参数估计并建立了估计量的相合性和渐近正态性，考虑了数值算法和假设检验问题，利用提出的方法研究了空气质量等级排名和市长公开电话均值突变检测等实际问题。在具有复杂网络结构时高维整数值GARCH模型的建模及其推断已完成初稿。