How exactly does the meaning of the Extreme Value Distribution differ from the distribution of the lowest/highest (extreme) order statistics?

I understand that the extreme value distribution (EVD) should apply for large sample sizes, but does that mean the extreme order statistic distribution should also converge to the EVD for large sample sizes?


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    $\begingroup$ For many distributions the extreme values will diverge as the sample size grows large; for others, they converge to constant values. The issue is similar to the Central Limit Theorem: you get convergence to a distribution only if you suitably standardize the statistic. $\endgroup$ – whuber Apr 25 '19 at 17:05

I'm not an expert on either, but as I understand Extreme Values they generally come in two flavors:

  1. Max (or min) in a given time period, and

  2. Values above a particular threshold (POT, Peaks Over Threshold)

Your alternative seems to be to look at all max values -- or some percentile -- which would still be different from #1 (which would aggregate to one maximum per time period), and would be more different from #2 since a threshold is an absolute number rather than percentile.

Not sure of the mathematical implications, though.

Per your comment, I believe that after the first step EV analysis then fits appropriate EV Distributions (Weibull, Gumbel, etc) to your data, which may or may not be the follow-on analysis that you're thinking of. And I'm not sure how much of the fitting depends on #1 or #2, above, as opposed to something close to that involving order statistics.

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  • $\begingroup$ Thanks, this makes some sense to me. I guess if you take the order statistic of sample size corresponding to the same time (return) period than that should agree with the EVD? $\endgroup$ – James Palmer Apr 25 '19 at 17:06
  • $\begingroup$ @JamesPalmer I've addressed your comment with what little additional EV info I know at the end of my posting. $\endgroup$ – Wayne Apr 25 '19 at 17:46

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