Distributional challenges regarding data on death and incidences during the SARS-CoV-2 pandemic up to July 2020

COVID-19 is a major global crisis with unpredictable consequences. Many scientists have struggled to make forecasts about its impact. Especially, appropriate preparations for a second wave are needed not to move in a costly panic mode again. It is necessary to get ideas about worst case scenarios regarding incidences, hospitalization, or use of ICU resources. They can be described in terms of extreme quantiles (95%, 99%, 99.9%) of specific distributions that supposedly formalize the data mechanism behind future observations. Therefore, distributional issues do matter. Cirillo and Taleb argue that a natural and empirically correct framework for assessing and managing real risk in pandemics is provided by extreme value theory dealing with extrema and not averages. We explore this idea in more detail. In this paper we discuss the fat-tail patterns in the distribution of the global COVID-19 data by analyzing data from 66 countries worldwide. We also explore their relevance at a lower, regional scale perspective (national, federal state), which is in our opinion more relevant for planning measures against the epidemic spread. For this we analyze data from the German federal state of Bavaria. We conclude that fat-tail patterns are seen in global data, possibly reflecting the respective heterogeneity between different countries regarding incidences and fatalities during the ongoing epidemic. However, the disease activity at regional level seems to be better described by classical Poisson based models. To bridge the gap between regional and global phenomena we refer to mixtures of slim-tail distributions that may create fat-tail features. Especially in the beginning of a pandemic acting according to the "better safe than sorry" principle and taking extreme forecasts as the basis for the decisions might be justified. However, as the pandemic continues and control measures are partially lifted, there is a need for a careful discussion how to choose relevant distributions and their respective quantiles for future resource planning in order not to cause more harm as the pandemic itself.