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).