Some Applications of Wavelets in Statistics

这个项目在研究和教育两个新方向上扩展了私人情报局目前的兴趣。该项目旨在开发小波方法及其在跨学科研究工作中的应用。自1989年Daubechies和Mallat发现正交小波基以来，小波作为P.I.S博士论文的主题，在许多科学领域得到了越来越广泛的应用。在接下来的五年里，P.I.计划在四个不同但又相互关联的领域中应用小波方法：i)非参数密度估计；ii)时间序列；iii)曲线回归中的降维；以及v)生物学中的跨学科研究。具体贡献包括基于非参数小波的假设检验的发展；随机过程参数的小波估计，重点是长记忆；贝叶斯小波分量选择技术，最后是小波方法在蛋白质和基因组分析中的应用。除了研究部分，P.I.还将开发一门关于小波方法的课程。理论将与许多实际感兴趣的领域的应用交织在一起，讲座将与计算机演示相结合。这门课程将针对统计和非统计专业的高级研究生。其目标将是培训学生，这些学生随后可能会完成关于小波的论文工作。本课程还将培养能够将小波方法应用于统计学以外的研究领域的学生。

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.