Some Problems in Spectral Methods and Discrete Probability

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In statistics, a fundamental goal of time series analysis is to estimate temporal structure in data. With the proliferation of such data as part of the big data revolution the need for more accurate, realistic, and computationally efficient methods is increasing. Periodic structure is key in processes of the natural world and in many man-made environments, and spectral analysis is the proper approach to estimate periodic, or near-periodic, structure in time series. Two areas in which important practical issues remain are time series regression and long range dependent time series. Standard regression models in statistics do not efficiently account for temporal structure while estimation of long range dependence can be unreliable in practical scenarios. In probability theory, a basic problem is to compute the probability of a finite union of events, which requires knowing the probability of the intersection of every subset of the events. Often, only single event and pairwise intersection event probabilities are known or can be computed efficiently. Therefore, tight and low complexity bounds using limited information are desirable. Considerable interest in this problem has persisted for over 50 years as applications of such bounds in system design and statistics has expanded. The proposed research will focus on designing and analyzing novel statistical procedures that meet contemporary challenges posed by time series regression and estimation of long range dependence, and on novel methodology for bounding a union probability. New and substantial research challenges arise when trying to estimate relevant temporal structure yet maintain interpretability of fitted parameters in many statistical regression contexts, when trying to estimate long range dependence accurately in the face of structural or extra-variation contamination in the time series, and when constructing bounds under complexity constraints.*** ***The research objectives are divided into three main themes: (1) The creation of tools to incorporate modern spectral methods into standard regression models, the improvement of robustness and flexibility of current frequency domain methods, and the statistical analysis of the new procedures; (2) The development of robust techniques to estimate long range dependence and the statistical analysis of these techniques; (3) The investigation of optimality of bounds and the construction and performance of low complexity suboptimal bounds under information constraints.******The training component of the proposed research will provide on average 2 M.Sc. and 3 Ph.D students each year with stimulating research challenges and immerse them in important current topics in statistics and probability. The research is expected to provide practical tools to increase the usefulness and practical application of time series regression models, to increase the applicability of long range dependent models, and to advance knowledge in an important problem in probability.*****