题 目：CONCEPT OF DENSITY FOR FUNCTIONAL DATA
报告人：Professor Peter G. Hall （The University of Melbourne）
时 间：2010年7月21日（周三）上午10:00
地 点：光华管理学院新楼217教室
摘要：The data in a sample of time series, for example graphs of average temperature or average rainfall at different weather stations, can be considered to be different realisations of the series. As for any dataset we can ask which realisations are more extreme; that is, what realisations lie in the tails, and what realisations lie towards the centre (for example, near the mode) of the distribution. Questions such as these raise the notion of probability density for time-series realisations, or for functional data. While it is possible to rank points in a function space in terms of their density within a ball of given nonzero radius, the conventional concept of a probability density function, constructed with respect to a ball of infinitesimal radius, is not well defined, not least because there is no natural analogue of Lebesgue measure in a function space. We suggest instead a transparent and meaningful surrogate for density, defined as the average value of the logarithms of the densities of the distributions of principal component scores, for a given dimension. This `density approximation' is readily estimable from data, and leads directly to estimators of the mode of a distribution of functions. In particular, the mode of a distribution of random functions is well defined, even if the density is not. Methodology for estimating densities of principal component scores is of independent interest; it reveals shape differences that have not previously been considered.
报告人简介：
Prof. Hall is currently a professor and ARC Federation Fellow at the Department of Mathematics and Statistics, University of Melbourne, and also has a joint appointment at University of California Davis. He previously held a professorship at the Centre for Mathematics and its Applications at the Australian National University.
Prof. Hall is among the world's most prolific and highly-cited authors in both probability and statistics. Mathscinet lists him with more than 500 publications as of January 2008. He has made very substantial and important contributions to nonparametric statistics, in particular for curve estimation and resampling: the bootstrap method, smoothing, density estimation, and bandwidth selection. He has worked on numerous applications across fields of economics, engineering, physical science and biological science. Prof. Hall has also made groundbreaking contributions to surface roughness measurement using fractals. In probability theory he has made many contributions to limit theory, spatial processes and stochastic geometry. His paper "Theoretical comparison of bootstrap confidence intervals" (Annals of Statistics, 1988) has been reprinted in the Breakthroughs in Statistics collection.
