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April 2013 On the conditional distributions of low-dimensional projections from high-dimensional data
Hannes Leeb
Ann. Statist. 41(2): 464-483 (April 2013). DOI: 10.1214/12-AOS1081

Abstract

We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random $d$-vector $Z$ that has a Lebesgue density and that is standardized so that $\mathbb{E} Z=0$ and $\mathbb{E} ZZ'=I_{d}$. Moreover, consider two projections defined by unit-vectors $\alpha$ and $\beta$, namely a response $y=\alpha'Z$ and an explanatory variable $x=\beta'Z$. It has long been known that the conditional mean of $y$ given $x$ is approximately linear in $x$, under some regularity conditions; cf. Hall and Li [Ann. Statist. 21 (1993) 867–889]. However, a corresponding result for the conditional variance has not been available so far. We here show that the conditional variance of $y$ given $x$ is approximately constant in $x$ (again, under some regularity conditions). These results hold uniformly in $\alpha$ and for most $\beta$’s, provided only that the dimension of $Z$ is large. In that sense, we see that most linear submodels of a high-dimensional overall model are approximately correct. Our findings provide new insights in a variety of modeling scenarios. We discuss several examples, including sliced inverse regression, sliced average variance estimation, generalized linear models under potential link violation, and sparse linear modeling.

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Hannes Leeb. "On the conditional distributions of low-dimensional projections from high-dimensional data." Ann. Statist. 41 (2) 464 - 483, April 2013. https://doi.org/10.1214/12-AOS1081

Information

Published: April 2013
First available in Project Euclid: 16 April 2013

zbMATH: 1360.62371
MathSciNet: MR3099110
Digital Object Identifier: 10.1214/12-AOS1081

Subjects:
Primary: 60F99
Secondary: 62H99

Keywords: Dimension reduction , high-dimensional models , regression , small sample size

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 2 • April 2013
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