How to build CDF when there are extreme values

Question

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).

Answer 1

Parametric families of distributions are generally very narrow classes of distributions (often varying only a few real parameters) and so they usually do not approximate broad classes of distributions very well. Consequently, if you have real data and you try to fit it to a parametric family of distributions, without some strong theoretical reason for thinking that the family will be a reasonable fit, you will often find that there is some aspect of the parametric family that does not adequately describe the real distribution of the sequence of data. You can certainly try to adapt things by trying different paraetric families of distributions, based on consideration of matters like tail fatness, etc., but you may find that there are still aspects of the chosen parametric family that don't fit well.

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.