Based on comprehensible non-parametric methods, estimates of crucial parameters that characterise the COVID-19 pandemic with a focus on the German epidemic are presented. Where appropriate, the estimates for Germany are compared with the results for six other countries (FR, IT, US, UK, ES, CH) to get an idea of the breadth of applicability and a relational understanding. Thereby, only prevalence data of daily reported new counts of diagnosed cases and fatalities provided by the ECDC are used. Where appropriate, the results are compared with conclusions drawn from using the dataset provided by the RKI. Drawing on uncertain a priori knowledge is avoided. Specifically, we present estimates for the duration from diagnosis to death being 13 days for Germany and about 2 days for Italy as the extremes. Furthermore, based on the knowledge of this time lag between diagnoses and deaths, properly delayed asymptotic as well as instantaneous fatality-case ratios are calculated having superiority compared to the commonly published case-fatality rate. The median of the time series of the instantaneous fatality-case ratio with proper delay of 13-days between cases and deaths for Germany turns out to be 0.024. Asymptotic values are presented for other countries with France ranking highest with a fatality-case ratio of almost 0.2 at its peak. The basic reproduction number, R_0, for Germany is estimated to be between 2.4 and 3.4. The uncertainty stems from uncertain knowledge of the generation time. A delay autocorrelation shows resonances at about 4 days and 7 days, where the latter resonance is at least partially attributable to the sampling process with weekly periodicity. The calculation of the basic reproduction number is based on an evaluation of cumulative numbers of cases yielding time-dependent doubling times as an intermediate step. This allows to infer to the reproduction number during the early phase of onset of the epidemic. In a second approach, the instantaneous basic reproduction number is derived from the incident (counts of new) cases and allows, in contrast to the first version, to infer to the temporal behaviour of the reproduction number during the later epidemic course. To conclude, by avoiding complicated parametric models we provide insights into basic features of the COVID-19 epidemic in an utmost transparent and comprehensible way.