High precipitation quantiles tend to rise with temperature,
following the so-called Clausius–Clapeyron (CC) scaling. It is often reported
that the CC-scaling relation breaks down and even reverts for very high
temperatures. In our study, we investigate this reversal using observational
climate data from 142 stations across Germany. One of the suggested
meteorological explanations for the breakdown is limited moisture supply.
Here we argue that, instead, it could simply originate from undersampling. As
rainfall frequency generally decreases with higher temperatures, rainfall
intensities as dictated by CC scaling are less likely to be recorded than for
moderate temperatures. Empirical quantiles are conventionally estimated from
order statistics via various forms of plotting position formulas. They have
in common that their largest representable return period is given by the
sample size. In small samples, high quantiles are underestimated accordingly.
The small-sample effect is weaker, or disappears completely, when using
parametric quantile estimates from a generalized Pareto distribution (GPD) fitted with

The atmospheric water holding capacity and thus potential
precipitation intensity depends exponentially on air temperature according to
the Clausius–Clapeyron (CC) relationship. As empirically documented by
several studies, high precipitation quantiles rise with temperature,
increasingly so with shorter duration, such as hourly or shorter. This
CC scaling describes a log-linear dependence of precipitation intensity on
temperature (

Regions and temperatures used in the literature cited in Fig

Several explanations for this phenomenon have been proposed, such as an
increase in the proportion of rainfall stemming from convective events as
opposed to large-scale stratiform precipitation

We analyzed publicly available time series of precipitation, temperature, and
relative humidity from 142 stations across Germany from the German Weather
Service

To analyze only the nonzero precipitation records that are actually of interest for this article, values below 0.5 mm h

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

Following the analysis method of

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

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

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

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
(

The GPD quantile computation formula used in the source code of lmomco is

In Sect.

We apply the results of the previous Sect.

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.

Median of the empirical and parametric 99.9 % quantile estimates depending on the size of
samples drawn from all the precipitation intensity values along with their uncertainty bands.
The horizontal dashed line marks the empirical quantile of the complete dataset (

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 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 99.9 % precipitation intensity per temperature bin with empirical and parametric quantile estimate
(

The synthetic

Precipitation quantile estimates rise with temperature until they reach a turning point, beyond which they decrease. For this drop in the CC-scaling relation towards higher temperatures, a number of explanations have been suggested. In this study we offer the alternative view that the drop can be understood, at least in some cases, as a statistical artifact of small samples. At higher temperatures, fewer precipitation observations are available because (1) wet events are less frequent at high temperatures and (2) precipitation events at higher temperatures are generally convective in nature and very localized in space; they are thus often missed by the observing network, resulting in smaller sample sizes compared to large-scale precipitation at lower temperatures. A rather simple argument shows that empirical quantile estimators have an underestimation bias for return periods exceeding the sample size, and we verified this behavior in a set of Monte Carlo experiments. It turned out that the underestimation of high quantiles, such as those relevant for the upper portion of the CC-scaling relationship, can be substantial. We have shown that when empirical estimators are appropriately replaced by parametric ones, the high-temperature drop in CC scaling disappears. The method of parametric estimation is crucial, nevertheless, as similar small-sample biases are known, e.g., from using MLE estimators (see above and more examples in the Supplement). The most robust estimates were obtained from moment-based methods. Past CC-scaling studies that have relied on empirical or ML-based quantile estimators are likely affected by the small-sample artifacts for high temperatures that we have described here. For those, we find it necessary to revisit the corresponding estimation step using other, e.g., moment-based, procedures. This may be especially interesting for quantiles beyond the 99.9 % level.

To exclude potential physical effects related to precipitation as much as
possible, we have repeated the analysis with synthetic data and obtained
essentially the same results. Furthermore, we have used dew-point temperature
instead of air temperature in order to rule out that the drop in the

Parametric quantiles from fitted distributions provide a means to retrieve less biased estimates of extreme quantiles. The price to be paid is the larger uncertainty of those estimates. This should be quantified by confidence intervals or application to several datasets to avoid singular non-representative results. The parametric method requires significantly fewer data points in a sample than empirical quantiles need to converge to the actual (unknown) value. In the combination of small sample sizes and very high quantiles, the use of parametric quantiles is recommended.

The datasets are freely available through the DWD
Climate Data Center. The complete analysis code and more graphical results
are available at

BB conducted the analysis and wrote the manuscript. GB and MH came up with the original idea and provided guidance and review.

The authors declare that they have no conflict of interest.

We wish to thank DWD for preparing and providing the datasets as well as William Asquith for reviewing our manuscript before submission. We are indebted to the reviewers for their many suggestions that led to the published version of this article. Edited by: Uwe Ulbrich Reviewed by: Reik Donner and two anonymous referees