Statistical Analysis of Time Series with Long Memory

在过去的几十年里，长记忆统计模型已经成为理论和应用时间序列分析的重要组成部分。 这些模型的特点是在无穷远处缓慢衰减的相关函数，或在原点具有奇点（极点或零点）的谱密度。 这些特征从根本上改变了统计估计和预测过程，因此，许多用于分析短记忆时间序列模型的方法和结果不再适用。 该提案的主要目标是为拥有各种类型的记忆结构的广泛的时间序列模型开发严格的估计和预测程序。 PI将研究二阶离散或连续时间平稳随机过程的估计和预测问题，其谱密度函数可能具有奇异性，以及Toeplitz算子理论的相关分析问题，这些问题在离散和连续时间情况下都可以作为工具。 在估计问题中，PI将研究模型未知参数的各种估计量的统计特性，这些特性取决于记忆结构和谱密度的平滑性。 在预测问题中，PI将研究相对预测误差随着观察过去的长度增加而减少的速率。在许多实际应用中（例如，经济学，金融学，计算机科学，水文学），数据是很好地描述了时间序列表现出长记忆性和季节性，或通过这样的时间序列的一些功能。 所提出的模型将所有这些作为特殊情况，并提供了一种合理的方法来分析某些宏观经济时间序列（通货膨胀和利率，货币总量，收入序列等），金融时间序列（金融资产收益的波动性、远期外汇市场溢价等）以及在计算机交通网络分析中出现的时间序列。

During the last decades, long memory statistical models have become an important part of theoretical and applied time series analysis. These models are characterized by slowly decaying correlation functions at infinity, or by spectral densities possessing singularities (poles or zeros) at the origin. These features change in an essential way the statistical estimation and prediction procedures, and as a consequence, many of the methods and results used for analyzing short-memory time series models are no longer appropriate. The main objective of the proposal is to develop rigorous estimation and prediction procedures for a broad class of time series models that possess various types of memory structures. The PIs will study estimation and prediction problems for second order discrete or continuous time stationary random processes with spectral density functions that may have singularities, and the related analytical problems from Toeplitz operators theory, which serve as tools both in the discrete and continuous time case. In the estimation problem, the PIs will investigate the statistical properties of various estimators of the unknown parameters of the model, which depend on the memory structure and the smoothness of the spectral density. In the prediction problem the PIs will study the rate of decrease of the relative prediction error as the length of observed past increases. In many practical applications (for instance, economics, finance, computer science, hydrology), the data is well described by time series exhibiting both long memory and seasonality, or by some functions of such time series. The proposed model captures all these as special cases, and provides a sensible way to analyze certain macroeconomic time series (inflation and interest rates, monetary aggregates, revenue series, etc.), financial time series (volatility of financial asset returns, forward exchange market premia, etc.) as well as time series which arise in the analysis of computer traffic networks.