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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, November 25, 2003}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Noureddine El-Karoui}
\centerline{\sl Stanford University}
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\centerline{\bf The Tracy-Widom law holds when $n, p, p/n \rightarrow \infty$, with application to PCA}
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Abstract:

Principal Component Analysis (PCA) is a tool used across the spectrum of
scientific applications. In modern practice, it is often applied to
$n\times p$ data matrices with $n$ and $p$ both large. Classical theory
(Anderson 1963) fails to apply in this setting. Using random matrix
theory, Johnstone (2000) recently shed light on some
theoretical aspects of PCA in this setup. Specifically, when the entries
of the $n\times p$ matrix $X$ are iid ${\cal N}(0,1)$ and
$n/p\rightarrow \rho \in (0,\infty)$, he showed that $\lambda_{n,p}$ ,
the largest eigenvalue of the empirical covariance matrix
$X'X$, converges to the so-called Tracy-Widom distribution (after proper
recentering and rescaling).

We will show that the result holds when $n,p\rightarrow \infty$ and 
$n/p\rightarrow 0$ or $\infty$, in effect removing the need to worry
about the limiting behavior of $n/p$. We will also present preliminary
results for rates of convergence. Finally, we will illustrate how these
and related theoretical insights might be used in practice.





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