Open Access
July, 1980 Canonical Variables as Optimal Predictors
V. J. Yohai, M. S. Garcia Ben
Ann. Statist. 8(4): 865-869 (July, 1980). DOI: 10.1214/aos/1176345079

Abstract

Let $\mathbf{X} = (X_1, \cdots, X_m)'$ and $\mathbf{Y} = (Y_1, \cdots, Y_n)'$ be two random vectors. Given any random vector $\mathbf{Z}$, let $\mathbf{Y}^\ast_Z$ be the best linear predictor of $\mathbf{Y}$ based on $\mathbf{Z}$. Let $p$ be any natural number smaller than $m$. We consider the problem of finding the $p$-dimensional random vector $\mathbf{Z} = (Z_1, \cdots, Z_p)'$ where each component $Z_i$ is a linear function of $\mathbf{X}$, which minimizes the determinant of $E(\mathbf{Y} - \mathbf{Y}^\ast_Z)(\mathbf{Y} - \mathbf{Y}^\ast_Z)'$. We show that $Z_1, \cdots, Z_p$ coincide with the first $p$ canonical variables (except for a nonsingular linear transformation). We also show that the square of the $(p + 1)$th canonical correlation coefficient measures the relative improvement in the prediction of $\mathbf{Y}$ when $p + 1 Z_i$'s are used instead of $p$.

Citation

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V. J. Yohai. M. S. Garcia Ben. "Canonical Variables as Optimal Predictors." Ann. Statist. 8 (4) 865 - 869, July, 1980. https://doi.org/10.1214/aos/1176345079

Information

Published: July, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0463.62054
MathSciNet: MR572630
Digital Object Identifier: 10.1214/aos/1176345079

Subjects:
Primary: 62H20

Keywords: canonical correlations , canonical variables , linear predictors

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 4 • July, 1980
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