## Electronic Communications in Probability

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

### Quantitative asymptotics of graphical projection pursuit

#### 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