Some Applications of Wavelets in Statistics

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This project extends current interests of the P.I. in new directions, in both research and education. The project aims at the development of wavelet methods as well as at their use in interdisciplinary research efforts. Wavelets, the P.I.'s Ph.D. thesis subject, have become increasingly popular in many scientific fields since the discovery of orthonormal wavelet bases byDaubechies and Mallat in 1989. The work that the P.I. intends to do over the next five years addresses the application of wavelet methods in four different but interrelated areas:i) nonparametric density estimation;ii) time series;iii) dimension reduction in curve regression, andiv) interdisciplinary research in biology.Specific contributions include the development of nonparametric wavelet-based hypothesis tests; the wavelet estimation of parameters of random processes, with emphasis on long-memory; Bayesian wavelet component selection techniques and, finally, applications of waveletmethods in the analysis of proteins and genomes.In addition to the research component, the P.I. will develop a course on wavelet methods. Theory will be interlaced with applications in many areas of practical interest and lectures integrated with computer demonstrations. The course will be at the advanced graduate level, for statisticsand non-statistics majors. The goal will be to train students that may subsequently do their dissertation work on wavelets. The course will also produce students capable of bringingwavelet methods in research fields other than statistics.