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April, 1995 Dimension of the Singular Sets of Plane-Fitters
Steven P. Ellis
Ann. Statist. 23(2): 490-501 (April, 1995). DOI: 10.1214/aos/1176324532


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.


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Steven P. Ellis. "Dimension of the Singular Sets of Plane-Fitters." Ann. Statist. 23 (2) 490 - 501, April, 1995.


Published: April, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0824.62052
MathSciNet: MR1332578
Digital Object Identifier: 10.1214/aos/1176324532

Primary: 62H99
Secondary: 62J99

Keywords: bootstrap , collinearity , Hausdorff dimension , jackknife , principal components , Projection pursuit , regression

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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