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 seven other countries (FR, IT, US, UK, ES, CH, BR) 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 Johns Hopkins University are used. Drawing on uncertain a priori knowledge is avoided. Specifically, we present estimates resulting from delay-time correlations for the duration from diagnosis to death being 13 days for Germany and Switzerland. The delay-time correlation applied to time series from other countries exhibit less pronounced peaks suggesting high variabilities for the corresponding time-to-death durations. With respect to the German data, the two time series of new cases and fatalities exhibit a strong coherence within the frequency range of interest, which backs our findings. 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 temporal median of the instantaneous fatality-case ratio with proper delay of 13-days between cases and deaths for Germany turns out to be 0.02. Time courses of asymptotic fatality-case ratios are presented for other countries which substantially differ during the first half of the pandemic, however, converge to a narrow range with standard deviation 0.57% and mean 2.4%. Additionally, the time courses of instantaneous fatality-case ratios with optimal delay for the 8 exemplarily chosen countries are calculated and compared by means of the temporal medians. Similarly to the asymptotic fatality-case ratios, the differences are much smaller than expected from mass media reports. The basic reproduction number, R₀, 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 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. The time course of the reproduction number is compared to an alternative control measure given by the per capita growth, which largely confirms the conclusions drawn from the reproduction number. 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. The perhaps most striking insight is that the fatality-case ratios do not differ between countries as much as previously suspected.