Computer-intensive Methods for Nonparametric Time Series Analysis

重采样和次采样提供了在依赖数据的背景下获得有效分布近似的可行方法，而对潜在的随机机制的假设很少。为了在实践中安全、准确地应用这些现代方法，仍然需要解决许多重要的问题。研究者希望解决的主要问题包括：(a)通过考虑自归一化统计量和/或可能有重尾的时间序列的极值，以及子抽样估计量的外推/内插，扩展子抽样的适用性领域。(b)研究局部Bootstrap在形成条件矩置信带和马尔可夫过程未来值预测区间以及构造时间可逆性假设检验方面的性能。(c)表明，当底层函数是超光滑时，通过平顶核平滑对条件矩的非参数估计不受维数诅咒的明显影响。(d)研究了新提出的局部块bootstrap在随机结构缓慢变化的非平稳序列中的性能。(e)提出了锥形块引导算法，并证明了与众所周知的块引导相比，它具有更好的性能。(f)提出一种新的具有优越收敛率的块/带宽选择估计器。最后(g)考虑可能集成时间序列的问题，并提出一种新的计算机密集型程序，连续路径块bootstrap，用于统计推断。相关数据，如时间序列和空间数据，经常在许多不同的科学学科中遇到，包括经济学，气象学，电气工程等。该项目的总目标是进一步发展计算机密集的统计分析方法，这些方法适用于相关数据的设置，但不依赖于不切实际或无法核实的模型假设。有效地解决这个问题将有许多实际应用。例如，在跨越十年（或更长时间）的每日汇率或股票回报序列中，可能有证据表明该序列的随机结构在如此长的时间内并非不变。创建一种实用的方法来模拟这种非平稳性，并设计适当的重新采样方法进行推理，将对经济应用最有帮助。对于不同的应用，考虑制造系统的随机仿真问题或吉布斯采样器仿真；为了评估模拟的收敛性和准确性，对“几乎”平稳时间序列的子采样/重采样的发展将是最有帮助的。在空间统计（例如采矿和地质统计、大气和环境科学等）方面，这些数据通常对应于在不规则间隔的空间点上获得的测量结果。例如，测量可以表明在某个地点X发现的矿石的质量或数量，或者在固定的时间间隔内测量地点Y的降水或空气质量。测量位置的不规则性质带来了额外的复杂性，然而，可以通过特殊设计的重采样/次采样版本来绕过。

Resampling and subsampling offer viable approaches to obtaining valid distributional approximations in the context of dependent data while assuming very little about the underlying stochastic mechanism. Many important questions still need to be addressed in order for these modern approaches to be applied safely and accurately in practice. The main issues the investigator wishes to tackle include the following: (a) Extend the realm of applicability of subsampling by considering self-normalized statistics and/or extrema of time series with possibly heavy tails together with extrapolation/interpolation of subsampling estimators. (b) Investigate the performance of the Local Bootstrap in forming confidence bands for conditional moments and prediction intervals for future values of a Markov process, as well as constructing hypothesis tests for time-reversibility. (c) Show that nonparametric estimation of conditional moments via flat-top kernel smoothing is not appreciably affected by the curse of dimensionality when the underlying function is ultra-smooth. (d) Investigate the performance of the newly proposed Local Block-Bootstrap in the case of a nonstationary series with a slowly-changing stochastic structure. (e) Propose the Tapered Block-Bootstrap algorithm, and show that it achieves superior performance as compared to the well-known Block-Bootstrap. (f) Propose a new block/bandwidth choice estimator with superior rate of convergence. And finally (g) consider the issue of a possibly integrated time series, and propose a new computer-intensive procedure, the Continuous-Path Block-Bootstrap, for statistical inference. Correlated data, such as time series and spatial data, are often encountered in many diverse scientific disciplines including economics, meteorology, electrical engineering, etc. The general goal of this project is to further the development of computer-intensive statistical analysis methods that are applicable in the setting of correlated data but do not rely on unrealistic or unverifiable model assumptions. Addressing this issue fruitfully will have many practical applications. For example, in a daily series of exchange rates or stock returns spanning a decade (or more), there may be evidence that the stochastic structure of the series has not been invariant over such a long stretch of time. Creating a practical way to model such nonstationarities and devising appropriate resampling methods for inference would be most helpful for economic applications. For a different application, consider the problem of stochastic simulation of manufacturing systems or a Gibbs-type sampler simulation; the development of subsampling/resampling for `almost' stationary time series would be most helpful in order to assess convergence and accuracy of the simulation. In the context of spatial statistics (e.g., mining and geostatistics, atmospheric and environmental science, etc.), the data typically correspond to measurements obtained at spatial points that are irregularly spaced. For example, a measurement may indicate the quality or quantity of the ore found in some location X, or a measurement of precipitation or air quality at location Y during a fixed time interval. The irregular nature of the measurement locations presents an added complication that, however, can be by-passed by specially designed versions of resampling/subsampling.