Short Memory in Long Memory Time Series

近30年来，长记忆过程的渐近性质和推理过程得到了广泛的研究。然而，当一个长记忆过程涉及短记忆成分时，统计推理方法是不够的，需要大量的改进。具体来说，如果一个分数积分的自回归移动平均（ARFIMA）过程包含自回归移动平均（ARMA）成分，目前应用的统计方法经常产生偏差，导致显著的不准确性。因此，有必要对ARFIMA过程中相关的短记忆组件进行更准确的调查。本项目考虑了时间序列设置中数据具有长记忆和短记忆特征的几个统计问题。考虑的统计问题包括：(1)检验长记忆时间序列是否具有短记忆特征；(2)建立了具有更简单自相关结构的长记忆和短记忆特征的随机参数回归模型；(3)评估具有短记忆特征的长记忆时间序列样本自相关和互相关的偏差。研究具有短记忆成分的长记忆时间序列非常重要，因为它们经常在现实环境中被观察到，例如股票回报和波动性、通货膨胀率、温度和河流水位。该项目旨在开发准确的统计模型和推理方法来分析这些时间序列。本研究的发展将：(1)促进长记忆过程的理论和方法的发展；(2)利用提出的模式和方法，帮助公众更好地了解全球变暖问题；(3)使从业人员在其学科中使用研究成果。此外，研究者将为博伊西州立大学数学和统计咨询中心的启动做出贡献。该中心将成为应用数学和统计学与其他科学融合的中心，为博伊西和爱达荷州提供服务，那里目前没有这样的设施。

Asymptotic properties and inference procedures for long memory processes have been extensively studied in the last 30 years. However, when a long memory process involves short memory components, statistical inference methods are insufficient and need to be substantially enhanced. Specifically, if a fractionally integrated autoregressive moving-average (ARFIMA) process contains autoregressive moving-average (ARMA) components, currently applied statistical methods frequently produce biases that result in significant inaccuracies. Accordingly, there is a need for a more accurate investigation of short memory components pertinent in the ARFIMA process. This project considers several statistical problems in time series settings where the data has both long memory and short memory characteristics. The statistical problems considered include: (1) testing to determine if a long memory time series has short memory characteristics; (2) developing stochastic parameter regression models of long memory and short memory characteristics with a simpler autocorrelation structure; and (3) assessing biases in the sample autocorrelations and cross-correlations for long memory time series with short memory characteristics.Studying long memory time series with short memory components is very important, as they are frequently observed in real-world contexts, such as stock returns and volatilities, inflation rates, temperatures, and river levels. The project aims to develop accurate statistical models and inference methods to analyze such time series. The development of this research will: (1) advance the theory and methods of long memory processes; (2) help the public better understand global warming issues with the proposed models and methods; and (3) benefit practitioners to use the research outcomes in their disciplines. In addition, the investigator will contribute to the launch of Boise State's mathematical and statistical consulting center. The center will be a hub of applied mathematics and statistics fused with other sciences, serving Boise and the State of Idaho where no such facility is currently available.