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One of the most serious shortcomings of covariance/correlation are the assumptions of linearity and normality.

What is the most natural generalization of these measures of dependence when you want to model the dependence structure of extreme events using heavy-tailed distributions, e.g. the Generalized extreme value distribution?

With "most natural generalization" I mean that the classical covariance/correlation is included as a special case when the usual assumptions hold.

(Disclosure: After having received no answers for nearly two weeks I posted this question also at Quantitative Finance)

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    $\begingroup$ I take it from your cross-post that you really want this question migrated. The effect of migration is to close the question on the original site and move it to the destination site. The move has been made; I'll take care of the close. (Interesting question, though: +1. I suspect it was simply overlooked here by many people. Our faq explains ways in which you can raise the visibility of a question that has languished.) $\endgroup$ – whuber Dec 21 '12 at 23:15
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I guess the answer really depends on what you want to do with it! It is tricky to measure association for extreme values, not the least because the correlation structure often will change in the extreme tails. Look at the stock exchange. Papers that in normal times do not correlate much, will tend to correlate more in time of crisis, since then all the papers move downwards.

Things you could look into: tail dependence coefficients, write into google "tail dependence coefficients" (without the "")

Extreme value copulas: Write into google "copula of extreme value distributions" (without the "")

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