# How to build CDF when there are extreme values

I want to build a CDF for some phenomenon, say $$P($$storm duration $$D \leq d)$$. The particularity of that phenomenon is that it has extreme values.

I understand that I can fit some PDF (I have data for that) to find my CDF. If I do it however, the PDF doesn't fit the extreme values very well. Reading the web, I see that one can fit Generalized Extrema Values, for example, fitting any duration above a threshold $$u$$.

Now I can model things under $$u$$ and above $$u$$, that is: $$P(D \leq x | x \leq u)$$ and $$P(D \leq x | x > u)$$. That is, I can build two CDF's. How can I join them into single one ? My understanding is that I can pick the values of the first one (regular fit) up to my threshold $$u$$ and then switch to the value of the second one (extreme values fit) when I'm above $$u$$. But doing that introduces a discontinuity in the CDF which I'm not sure is acceptable.

Is my understanding correct ? Does it lead to good approximation of the "real" CDF ? How do I handle the discontinuity ?

After comment: the extreme values are located in the right tail of the real distribution. In my case, storms which last a few hours (regular case) are much more frequent than those which last for days (extreme case).

• Are there 'extreme values' in both tauk? Id ao Hve considered that your data may be Mar 20, 2022 at 23:19

If you have a sequence of IID data, and you would like to fit your data to a distribution that will converge to the true distribution in the limit, you might be better off considering some kind of kernel density estimator, using a kernel distribution that has fat enough tails to reflect the extreme values you want to model. For example, as a reasonable starting point you could try representing the distribution of the log-duration using a KDE with kernels given by the T-distribution with degrees-of-freedom $$1 \leqslant v \leqslant 2$$ (depending on how fat you want the tails to be). If you have a reasonable amount of data then you should get a reasonably smooth distribution estimator, and the right tail will be quite fat, which should adequately encapsulate your expectation that there will be extreme values in the distribution.
• Although this is an interesting answer, it doesn't answer the actual question which is how to "merge" two CDF, knowing one is "valid" from 0 to $u$ and the other from $u$ to some maximum. Mar 22, 2022 at 13:41