The modified Yule-Walker method for multidimensional infinite-variance periodic autoregressive model of order 1
多维无限方差一阶周期自回归模型的修正 Yule–Walker 方法AI 解读 ↓
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多维无限方差一阶周期自回归模型的修正 Yule–Walker 方法
- 研究问题
- 当多维周期自回归序列的噪声服从无限方差的 α 稳定分布、通常的协方差不存在时，如何估计其模型参数？
- 方法与思路
- 作者以协变差替代 Yule–Walker 方程中的协方差，分别利用矩表示与谱测度表示估计协变差，并开展模拟和真实数据分析。
- 材料陈述的贡献
- 摘要提出适用于多维 α 稳定周期自回归模型的新估计方法，并比较两种协变差估计途径在不同样本量和稳定指数下的表现。
- 阅读关注点
- 应核对协变差的定义与存在条件、两种估计量的实现要求，以及周期结构、稳定指数和样本量如何影响参数估计表现。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未经核实；摘要未说明稳定指数的适用范围、理论保证及真实数据分析的具体结果。
原文摘要
The time series with periodic behavior, such as the periodic autoregressive (PAR) models belonging to the class of the periodically correlated processes, are present in various real applications. In the literature, such processes were considered in different directions, especially with the Gaussian-distributed noise. However, in most of the applications, the assumption of the finite-variance distribution seems to be too simplified. Thus, one can consider the extensions of the classical PAR model where the non-Gaussian distribution is applied. In particular, the Gaussian distribution can be replaced by the infinite-variance distribution, e.g. by the $α-$stable distribution. In this paper, we focus on the multidimensional $α-$stable PAR time series models. For such models, we propose a new estimation method based on the Yule-Walker equations. However, since for the infinite-variance case the covariance does not exist, thus it is replaced by another measure, namely the covariation. In this paper we propose to apply two estimators of the covariation measure. The first one is based on moment representation (moment-based) while the second one - on the spectral measure representation (spectral-based). The validity of the new approaches are verified using the Monte Carlo simulations in different contexts, including the sample size and the index of stability of the noise. Moreover, we compare the moment-based covariation-based method with spectral-based covariation-based technique. Finally, the real data analysis is presented.