Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 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
  1. Persi Diaconis and David Freedman. Asymptotics of graphical projection pursuit. Ann. Statis., 12(3):793--815, 1984. Math. Review MR751274
  2. A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large matrices. Electron. Comm. Probab., 5:119--136 (electronic), 2000. Math. Review MR1781846
  3. Elizabeth Meckes. An infinitesimal version of Stein's method of exchangeable pairs. Doctoral dissertation, Stanford University, 2006. Math. Review number not available.
  4. Elizabeth Meckes. Linear functions on the classical matrix groups. Trans. Amer. Math. Soc., 360(10):5355--5366, 2008. Math. Review MR2415077
  5. Vitali D. Milman and Gideon Schechtman. Asymptotic Theory of Finite-dimensional Normed Spaces, volume 1200 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. Math. Review MR856576
  6. Charles Stein. The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report No. 470, Stanford University Department of Statistics, 1995. Math. Review number not available.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X