Electronic Communications in Probability

Quantitative asymptotics of graphical projection pursuit

Elizabeth Meckes

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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 $\{x_i\}$ in $R^d$ 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 $x_i$ 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 $x_i$, if directions are tried independently and uniformly.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 17, 176-185.

Dates
Accepted: 3 May 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234726

Digital Object Identifier
doi:10.1214/ECP.v14-1457

Mathematical Reviews number (MathSciNet)
MR2505173

Zentralblatt MATH identifier
1189.60046

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E20: Asymptotic distribution theory

Keywords
Projection pursuit concentration inequalities Stein's method Lipschitz distance

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Meckes, Elizabeth. Quantitative asymptotics of graphical projection pursuit. Electron. Commun. Probab. 14 (2009), paper no. 17, 176--185. doi:10.1214/ECP.v14-1457. https://projecteuclid.org/euclid.ecp/1465234726


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References

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