The statistical analysis of time series and random fields is vital in many diverse scientific disciplines. This project continues the development of computer-intensive statistical methods of inference for the analysis of dependent data without relying on unrealistic or unverifiable model assumptions. In particular: (a) General estimators are constructed based on nested subsample values of converging/diverging statistics with general applications including tail index and rate estimation. (b) Limit theorems are proven for the distribution of self-normalized statistics from marked point processes with (possibly) heavy tails; it is shown how subsampling can be used for inference purposes without explicit knowledge and/or estimation of the heavy tail index. (c) It is demonstrated that the use of special `flat-top' kernels is advised both in the context of residual bootstrap, as well as in the problems of functional estimation in nonparametric autoregression, estimation of conditional moments, and spectral density and large-sample covariance matrix estimation. (d) Two different block bootstrap schemes, one based on a local blocking technique and the other on residuals, are devised to address data from locally (but not globally) stationary series. (e) The way to conduct a most powerful bootstrap hypothesis test in linear/nonlinear (auto)regression set-ups is identified and powerful bootstrap unit root tests are devised as a result.
This project falls in the realm of nonparametric statistics where inferences (estimation, confidence intervals, hypothesis tests, etc.) are carried out without relying on ad hoc model assumptions. In some sense, the nonparametric viewpoint allows the data to ``speak for itself'', and is particularly appropriate in a `large-sample' situation where data are abundant; in our information-explosion age, this is progressively a typical situation. For example, in a daily series of exchange rates or stock returns spanning a decade, or a series of (average) annual temperatures over the last 100 years, there may be evidence that the stochastic structure of the series has not remained invariant over such a long stretch of time. Part of this project deals with devising appropriate computer-intensive methods for inference in such nonstationary environments (e.g., trend detection and estimation) that would be most helpful in economic applications as well as the problem of climate change. As another example, consider meteorological data gathered from weather stations scattered all around the country; since the spatial locations of the measurements are highly irregular, this type of data constitutes a so-called `marked point process'. The work under this project provides powerful methodology for the analysis of data under such practically important and difficult settings.
