The Annals of Statistics

Dimension of the Singular Sets of Plane-Fitters

Steven P. Ellis

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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

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

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H99: None of the above, but in this section
Secondary: 62J99: None of the above, but in this section

Bootstrap collinearity Hausdorff dimension jackknife principal components projection pursuit regression


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

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