On a matrix‐valued autoregressive model

Many data sets in biology, medicine, and other biostatistical areas deal with matrix‐valued time series. The case of a single univariate time series is very well developed in the literature; and single multi‐variate series (i.e., vector time series) though less well studied have also been developed. A class of matrix time series models is introduced for dealing with situations where there are multiple sets of multi‐variate time series data. Explicit expressions for a matrix autoregressive model along with its cross‐autocorrelation functions are derived. Stationarity conditions are also provided. Least squares estimators and maximum likelihood estimators of the model parameters and their asymptotic properties are derived. Results are illustrated through simulation studies and a real data application.

生物学、医学和其他生物统计学领域的许多数据集都涉及矩阵值时间序列。单一单变量时间序列的情况在文献中已有充分研究；单一多元序列（即向量时间序列）虽然研究较少，但也有所发展。引入了一类矩阵时间序列模型，用于处理存在多组多元时间序列数据的情况。推导了矩阵自回归模型及其交叉自相关函数的显式表达式。还提供了平稳性条件。推导了模型参数的最小二乘估计量和最大似然估计量及其渐近性质。通过模拟研究和实际数据应用对结果进行了说明。