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I've always believed that the Spearman rank correlation matrix doesn't need to be positive semidefinite because the correlations are estimated pairwise so there's always a chance that it may not be. Recently, however, I have been looking deeper into this issue and I am confused, particularly after reading this:

https://www.usenix.org/legacy/event/sysml07/tech/full_papers/sabato/sabato.pdf

On page #4, Theorem 2 it says:

A Spearman rank correlation matrix is PSD

Proof: A Spearman correlation is a Pearson correlation applied to ranks. Therefore the Spearman rank correlation matrix is PSD

So... can the Spearman correlation matrix be not positive semidefinite? Is there a counter-example people know? Or is it a misunderstanding to think that the Spearman correlation matrix could be non-positive definite?

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A further lit search clarified this on here:

https://www.econ.kuleuven.be/public/ndbae06/PDF-FILES/grcort.pdf

Page #13 explains that the lack of positive definiteness comes from trying to transform the Spearman (or Kendall) correlation coefficient back into Pearson correlations by using the r = 2sin(pi/6*rho) formula (in which case the transformation needs to be done one correlation coefficient at a time, where the lack of positive definiteness may arise). As it is (i.e. the Pearson correlation on the ranked variables) the Spearman correlation matrix seems to always be positive (semi) definite.

My bad! Sorry! :)

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