Mathematical Sciences: Selected Large Sample Theory Problemsin Probability and Statistics

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This research involves work in four areas of probability and statistics, the bootstrap, large deviations, random fractals, and empirical processes indexed by sets. Research in the bootstrap area will involve investigation of the large sample properties of a general weighted bootstrap, which includes the nonparametric and Bayesian bootstraps as special cases. The research in large deviations will be centered around a notion of extended functional large deviations for partial sums. Random fractals will be studied in empirical increments of various random processes. The behavior of tail empirical processes indexed by sets will be studied with an eye towards applications, the most prominent being in establishing the consistency and asymptotic distribution of various multivariate density and mode estimators. Probability theory provides a basis for research in statistics. This research will include both the development of new results in probability, as well as their application to statistical theory.