Approximate and Exact Inference Via Computer-Intensive Methods

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The investigator will continue the development of inferential methods that do not rely on unrealistic or unverifiable model assumptions. Standard inferential methods rest upon strong assumptions, especially in the analysis of time series, random fields, or whenever complex dependencies must be taken into account. In contrast, resampling, subsampling, and other computer-intensive methods offer viable approaches to obtaining valid distributional approximations while assuming very little about the stochastic mechanism generating the data. In part 1 of this proposal, the investigator will address several important problems so that these bootstrap and subsampling methods can serve as good approximate methods in statistical practice. The main issues we wish to tackle include the following: further relaxing of conditions (such as slower mixing rate for long memory data and allowing for nonstationarity); more theory for irregularly spaced data; studying delicate problems where the rate of convergence depends on unknown parameters, such as in the notoriously difficult problem of autoregressive type processes with unit roots; improve the accuracy of distribution estimation by techniques such as Richardson extrapolation, and by optimal choice of block size; and pursue the development of goodness-of-fit tests in the dependent data case. These methods are especially useful in modelling of economic time series, due to the inherent difficulties caused by nonlinearity and nonstationarity. In part 2 of the proposal, the investigator will pursue the development of methods that have exact finite sample validity, such as the construction of conservative confidence regions, without the expense of losing efficiency, at least in large samples. Typical nonparametric methods are based on approximations or limit theorems, so that finite sample behavior is always an issue, and is often typically addressed by small scale simulations. In contrast, the goal here is to construct nonparametric procedures with guaranteed finite sample behavior and good efficiency.The statistical analysis of data is vital in many diverse scientific disciplines: physics, engineering, acoustics, geostatistics, medicine, econometrics, seismology, law, ecology, and others. The scope of modern statistical analysis is continually expanding, as is the need for inferential methods that are valid without imposing strong model assumptions. The investigator will continue the pursuit of the development of statistical methods that can be applied safely in practice, keeping in mind the many applications toward which such methods can fruitfully be applied. The philosophical approach of the investigator is to develop practical methods that have a robustness of validity so that they may be applied in increasingly complex situations. The impact of this work is potentially quite large because strong inferential statements can be made without imposing strong assumptions.