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Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present asymptotic results like "the distribution of components converges to normal", but I haven't found any work describing convergence rates or other properties of the approximation in detail.

In particular, I've run a number of numerical experiments with differently distributed data which seem to indicate that components contributing only a small amount of variance gravitate towards some platykurtic distribution with parameters largely independent of the inital distribution - even in the finite case. Does anyone have a hint where I might look for helpful results?

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A relatively well-known null model for PCA (at least for eigenvalue distributions) seems to be Tracy-Widom. In general, I think that "what patterns you would expect for PCA of random data" is still under active research as part of random matrix theory.

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  • $\begingroup$ Thank you! I know about Wigner's semi-circle-law and Tracy-Widom, but unfortunately they don't help in my case, as the time-series I'm concerned with show strong auto- and cross-correlations, so the distribution of the eigenvalues is quite off, anyway. $\endgroup$
    – Alexander Klein
    Commented Feb 27, 2013 at 8:59
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Try a multivariate statistics book. For eg. T.W. Anderson Intro to Multivariate Statistics. The joint distribution of the eigenvalues will be available under normality assumptions. That is a good place to start. Tracy-Widom is the distribution of the largest eigenvalue. Sometimes not very useful.

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