We analyzed publicly available time series of precipitation, temperature, and relative humidity from 142 stations across Germany from the German Weather Service . The stations are selected based on the length of available hourly time series. All selected datasets contain at least 15 years of observations, mostly 20 years. The R code for data selection, download, and analysis is available at https://github.com/brry/prectemp.

To analyze only the nonzero precipitation records that are actually of interest for this article, values below 0.5 mm h-1 are omitted. This cutoff is in line with the cited literature and is suitable because measurements of very low rainfall intensities have a high relative uncertainty. The values are then logarithmized to enable a comparison of rates of precipitation change across temperatures. Because of the very skewed nature of rainfall values, this also allows for better distribution fits.

Throughout this paper, event dew-point temperature is used as an integrated measure of air temperature and water vapor saturation (or moisture supply). It is defined as the average dew-point temperature of the 5 h preceding each rainfall hour, similar to the procedure by . Dew-point temperature is calculated with the Magnus formula based on observed relative humidity and air temperature at 2 m height see.

Following the analysis method of and , we partition the hourly precipitation depths according to the event dew-point temperature. We use moving temperature bins with a fixed width of 2 K. Bin midpoints increase in 0.1∘ steps.

Empirical quantiles are estimated by a monotonic mapping of the ordered sample to sample-size-specific probabilities called plotting positions. This can be done in a variety of ways as reviewed by . Common to all is the fact that the portion to the right of the sample maximum is left unresolved (no extrapolations) and receives the same probability as the maximum. Quantiles representing return periods larger than the sample length are consequently mapped to that maximum. They are therefore underestimated – a fact apparently too trivial to have warranted any publication. The empirical quantiles used in this article are computed based on the k-1/3n+1/3 plotting positions n= sample size, k = 1,..., n; see.

The parametric quantile estimates are obtained in a peak-over-threshold approach, where the generalized Pareto distribution is fitted to the top 10 % of the sample. Quantiles are calculated from the fitted GPD.

We use the method of L moments to fit the GPD parameters. They are analogous to the conventional statistical moments (mean, variance, skewness, and kurtosis) but “robust [and] suitable for analysis of rare events of non-normal data. L moments are consistent and often have smaller sampling variances than maximum likelihood in small to moderate sample sizes. L moments are especially useful in the context of quantile functions” .

To obtain the quantile from the fitted distribution, the given probabilities must be scaled with the conditional probability of the truncation. For example, if the 99 % quantile (Q0.99) is to be computed from the top 10 % of the data, Q0.90 of the truncated sample must be used. We refer to Q0.99 as the “censored 99 % quantile”. Because five values are required to obtain L moments, the minimum sample size at 90 % truncation is 50 (45 values are discarded).

Selecting a suitable fitting method is of great importance in the context of sample size bias. For example, unlike moment-based procedures, maximum likelihood estimation (MLE) can still show an underestimation bias at small sample sizes, as shown in the Supplement. This happens in small samples (n<200) for distributions with bounded parameters (and the optimum of the likelihood function lying on the boundary). We refer to the Supplement for a comparison of the different methods.

The GPD quantile computation formula used in the source code of lmomco is x(F)=ξ+ακ1-(1-F)κif κ≠0ξ-α×log⁡(1-F)if κ=0,withξ=location,α=scale,κ=shape.

In Sect. , we pointed out that empirical methods inherently underestimate high quantiles in small samples. In order to quantify the potential effect in the context of P–T relationships, we set up the following experiment: to investigate the dependency of both quantile estimation methods on sample size, we draw random samples from a defined population. This should optimally be a large set of values following a distribution observed in nature. We therefore use a pooled dataset with all the precipitation values observed at any of the 142 stations. From this population, we draw random samples of several sizes and compute empirical and parametric quantiles from each sample. For each sample size, this is done 1000 times, resulting in a corresponding quantile distribution depending on sample size.

We apply the results of the previous Sect.  – that is, the potential small-sample effects of empirical and parametric quantile estimates – to P–T scaling relationships and analyze the drop at high temperatures. To study that effect, we designed an experiment with synthetic data. Here, precipitation values are generated in a way that exhibits a stable temperature scaling over all temperature ranges. The CC-scaling rate is constant, and the increase in high rainfall quantiles per degree Kelvin remains the same over all temperatures. When sampling from such synthetic data, any drop in the P–T relationship must be a statistical artifact. For this purpose, we define a “temperature-dependent GPD” with parameters that depend on temperature. To achieve a realistic temperature scaling, we base the parameters on the linear regression of the fitted parameters at several dew-point temperatures.

From that synthetic GPD, 1000 random samples are generated for each temperature bin. The sample size corresponds to the average number of precipitation observations at the climate stations in each bin. From these sets of random samples, the empirical and parametric 99.9 % quantiles are calculated.

The dependence on sample size, as revealed by 1000 random draws per sample size from the pooled precipitation data, is shown in Fig. . The 99.9 % quantile of this population (n=1.16 million) is 19.5 mm h-1. It is strongly and consistently underestimated by the empirical estimator with shrinking sample size. For a sample size of 50, the median estimate is only 7 mm h-1. Realistic estimates are obtained only for samples larger than about 700, around which the estimates converge to the (true) population value. The parametric estimators do not exhibit this bias – only their variance increases with smaller samples (the uncertainty range is wider). This is a typical example of the well-known bias–variance tradeoff in estimation theory.

The procedure of obtaining parametric (using the GPD) and empirical quantiles was applied per temperature bin to the datasets of each of the 142 stations. The empirical precipitation quantiles per bin are presented in the left panel of Fig. . The shape of the P–T relationships is consistent with the behavior of P–T relationships shown in Fig.  of the introductory section. The empirical quantile estimates start decreasing between 15 and 20 ∘C. Some stations show the empirical quantile drop more distinctly than others. The figure also shows the average across stations, where the drop becomes particularly clear. Compared to the red line depicting the CC scaling of 7 to 6 % K-1, the precipitation increase follows a super-CC scaling with a rise that is steeper than the CC rate. This is in accordance with previous findings, e.g., by .

The parametric estimates are displayed in the right panel. At temperature ranges where empirical quantiles decrease, parametric quantiles keep increasing. This difference is less pronounced for smaller quantiles (see Supplement Sect. S4).

The synthetic P–T relationship that continuously rises with temperature (see Sect. ) is defined with the parameters shown in the left panels of Fig. , where each dot represents one of the stations. The right panel shows the median of the 99.9 % quantile estimates from random samples with the original sample sizes. Even though the distribution continues to increase with temperature, empirical quantiles from random samples stagnate or drop around 18 ∘C where sample size decreases quickly. Parametric quantiles obtained by distribution fitting do not drop and follow the theoretical quantile from the distribution function.