Computer-intensive Methods for Nonparametric Time Series Analysis

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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.