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\footnotesize \noindent Statistica Sinica {\bf 19} (2009), 1557-1565

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\normalsize
 \centerline{\large\bf ON PRINCIPAL COMPONENTS AND}
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  \centerline{\large\bf REGRESSION: A STATISTICAL EXPLANATION}
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 \centerline{\large\bf OF A NATURAL PHENOMENON}
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 \author{Andreas Artemiou and Bing Li}
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 \centerline{\it Pennsylvania State University}
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\begin{quotation}
\noindent {\it Abstract:} In this note we give a probabilistic
explanation of a phenomenon that is frequently observed but whose
reason is not well understood. That is, in a regression setting, the
response ($Y$) is often highly correlated with the leading principal
components of the predictor ($\Xb$) even though there seems no
logical reason for this connection. This phenomenon has long been
noticed and discussed in the literature, and  has received renewed
interest recently because of the need for regressing $Y$ on $\Xb$ of
very high dimension, often with comparatively few sampling units, in
which case it seems natural to regress on the first few principal
components of $\Xb$. This work stems from a discussion of a recent
paper by Cook (2007) which, along with other developments, described
a historical debate surrounding, and current interest in, this
phenomenon. \par
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 \noindent
{\it Key words and phrases:} Dimension reduction, orientationally
uniform distribution, \hskip 3pt principal components, random
covariance matrices, regression, stochastic ordering.
\par
\end{quotation}

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