Second Order Inference for Nonstationary Time Series

本文研究因果非平稳过程的二阶推理。虽然因果平稳过程可以被看作是通过过滤一组过去的新息而产生的，但人们可以允许过滤器是时变的，并且从此引入非平稳性。考虑两组问题。首先，对于具有非平稳误差的线性模型，研究者讨论了最小二乘估计的协方差矩阵的估计，以及一般的M-估计。其次，PI研究观测时间序列的时变协方差函数、时变谱和协方差矩阵的估计。自协方差函数和谱的同时推断可以用来研究它们的模式和趋势，也是令人感兴趣的。该研究需要为非平稳过程开发几种工具，包括经验过程，高斯近似，强不变性原理和二次型的大偏差。平稳性在经典时间序列分析中发挥了重要作用，基本上说，整体结构不随时间变化。然而，在许多科学领域，包括经济学、工程学、环境科学、金融学和神经科学等，认为观测到的时间序列是平稳的是不现实的。研究结果将有助于理解不同学科数据的性质，做出预测和结论，并且建议中的二阶推理是普遍和基本的，将有助于进一步对非平稳时间序列进行统计分析。

The proposed research is on second order inferences for causal nonstationary processes. While a causal stationary process can be viewed as generated by filtering a set of past innovations, one can allow the filter to be time-changing, and henceforth introduce nonstationarity. Two sets of problems are considered. First, for linear models with nonstationary errors, the investigator addresses the estimation of covariance matrices of the least square estimates, as well as general M-estimates. Second, the PI studies the estimation of time-varying covariance functions, time-varying spectrum and covariance matrices of the observed time series. Simultaneous inferences of autocovariance functions and spectra can be used to study their patterns and trends, and are also of interests. The study requires several tools to be developed for nonstationary processes, including empirical processes, Gaussian approximations, strong invariance principles and large deviations for quadratic forms.Stationarity has played an important role in classical time series analysis, which basically says that the overall structure does not change over time. However, in many scientific fields, including economics, engineering, environmental science, finance, and neuroscience etc, it is not realistic to believe the observed time series are stationary. Results from the proposed research will be useful in understanding the nature of the data from various disciplines, making forecasts and conclusions.Furthermore, the second order inferences in the proposal are general and fundamental, and will facilitate further statistical analysis of nonstationary time series.