## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 2 (1995), 490-501.

### Dimension of the Singular Sets of Plane-Fitters

#### Abstract

Let $n > p > k > 0$ be integers. Let $\delta$ be any technique for fitting $k$-planes to $p$-variate data sets of size $n$, for example, linear regression, principal components or projection pursuit. Let $\mathscr{Y}$ be the set of data sets which are (1) singularities of $\delta$, that is, near them $\delta$ is unstable (for example, collinear data sets are singularities of least squares regression) and (2) nondegenerate, that is, their rank, after centering, is at least $k$. It is shown that the Hausdorff dimension, $\dim_H(\mathscr{Y})$, of $\mathscr{Y}$ is at least $nk + (k + 1)(p - k) - 1$. This bound is tight. Under hypotheses satisfied by some projection pursuits (including principal components), $\dim_H(\mathscr{Y}) \geq np - 2$, that is, once singularity is taken into account, only two degrees of freedom remain in the problem! These results have implications for multivariate data description, resistant plane-fitting and jackknifing and bootstrapping plane-fitting.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 2 (1995), 490-501.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324532

**Digital Object Identifier**

doi:10.1214/aos/1176324532

**Mathematical Reviews number (MathSciNet)**

MR1332578

**Zentralblatt MATH identifier**

0824.62052

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H99: None of the above, but in this section

Secondary: 62J99: None of the above, but in this section

**Keywords**

Bootstrap collinearity Hausdorff dimension jackknife principal components projection pursuit regression

#### Citation

Ellis, Steven P. Dimension of the Singular Sets of Plane-Fitters. Ann. Statist. 23 (1995), no. 2, 490--501. doi:10.1214/aos/1176324532. https://projecteuclid.org/euclid.aos/1176324532