# Extreme value theory for count data

I am aware of extreme value theory for continuous distributions. I need to fit an extreme value distribution to the maximum observation of number of events on a day, per month. This seems to be the block maxima problem, which is approximated by the GEV family of distributions for continuous distributions. How do I do this for count data?

As a secondary question, let's assume the basic count process is ~ Poisson. Then does this lead to a different answer to the original question?

I don't know a definitive answer for your primary question. Although I found the following two references:

Anderson, C. W., “Extreme value theory for a class of discrete distributions with applications to some stochastic processes”, Journal of Applied Probability, vol 7, 1970, pp. 99–113.

Anderson, C. W., “Local limit theorems for the maxima of discrete random variables”, Mathematical Proceedings of the Cambridge Philosophical Society, vol 88, 1980, pp. 161– 165.

For your secondary question, the CDF of the Poisson is $\frac{\Gamma(\lfloor k+1\rfloor,\lambda)}{\lfloor k\rfloor!}$ so $P(\max\limits_N X_n \leq M) = (\frac{\Gamma(\lfloor k+1\rfloor,\lambda)}{\lfloor k\rfloor!})^N$. Apply the difference operator (lag1) and you get the PMF of the max.

For any recent visitors, there's been new developments in this area by Hitz, Davis and Samorodnitsky (arXiv:1707.05033). Taking a peaks-over-threshold approach instead of block maxima, the Discrete Generalised Pareto Distribution is derived as the $$\operatorname{floor}$$ of a GPD, and discrete Maximum Domains of Attraction (DMDA) are introduced by relating them to the classical MDAs. The whole thing is linked to, but different from, Zipf's Law.

In terms of the paper's terminology, the Poisson distribution is in the DMDA of a Gumbel distribution $$(\xi = 0)$$, as are the Negative Binomial and Geometric distributions.

• This looks more like a comment. Can you please make it more complete, by a short summary of what is in that paper? – kjetil b halvorsen Oct 21 '20 at 20:20
• @kjetilbhalvorsen I've edited the question, is that better? – Maximilian Aigner Oct 21 '20 at 20:33
• Much better, thanks! – kjetil b halvorsen Oct 21 '20 at 21:36

The Poisson does not fall within the MDA of any EV distribution (not possible to find shifting and scaling sequences that provide a non-degenerate limit). Consequence of a theorem in Leadbetter, Lindgren and Rootzen (1983).