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 Electronic Communications in Probability > Vol. 14 (2009) > Paper 17 open journal systems 


Quantitative asymptotics of graphical projection pursuit

Elizabeth S Meckes, Case Western Reserve University


Abstract
There is a result of Diaconis and Freedman which says that, in a limiting sense, for large collections of high-dimensional data most one-dimensional projections of the data are approximately Gaussian. This paper gives quantitative versions of that result. For a set of n deterministic vectors {xi} in Rd with n and d fixed, let &theta be a random point of the sphere and let &mu&theta denote the random measure which puts equal mass at the projections of each of the xi onto the direction &theta. For a fixed bounded Lipschitz test function f, an explicit bound is derived for the probability that the integrals of f with respect to &mu&theta and with respect to a suitable Gaussian distribution differ by more than &epsilon. A bound is also given for the probability that the bounded-Lipschitz distance between these two measures differs by more than &epsilon, which yields a lower bound on the waiting time to finding a non-Gaussian projection of the xi, if directions are tried independently and uniformly.


Full text: PDF

Pages: 176-185

Published on: May 3, 2009


Bibliography
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Electronic Communications in Probability. ISSN: 1083-589X