The study, led by Abubakar Haruna of the University Grenoble Alpes together with Juliette Blanchet, Guillaume Evin, and Emmanuel Paquet of EDF-DTG, was published in the journal Advances in Statistical Climatology, Meteorology and Oceanography. The researchers set out to answer a deceptively simple question: which probability distribution best models trends in low, medium, and extreme daily precipitation simultaneously? The answer matters far beyond academic statistics. Flood defenses, dam operations, agricultural planning, and urban drainage systems all depend on reliable estimates of how rainfall extremes are evolving, and a model that gets the tails of the distribution wrong can dangerously underestimate future flood risk.

Daily precipitation is notoriously awkward to model statistically. It is bounded below at zero, strongly skewed, and interrupted by dry days that produce a spike of probability mass at exactly zero rainfall. Traditional approaches often sidestep this complexity by analyzing only the mean, or by treating extremes separately using extreme value theory, for example with the generalized extreme value distribution fitted to annual maxima or the generalized Pareto distribution applied to values above a high threshold. But such split approaches introduce artificial discontinuities: the bulk of the distribution and its tail are described by different models that need not be statistically consistent with one another. For applications like stochastic rainfall simulation, where the full distribution of precipitation amounts is required, that inconsistency is a serious liability.

To avoid these pitfalls, the team adopted a mixed-type distribution that handles the discrete and continuous nature of rainfall in one coherent framework. A logistic model, driven by time as a covariate, describes the probability of a dry day, while a parametric distribution describes the intensity of rainfall on wet days. The researchers identified four conditions an ideal wet-day distribution must satisfy: it must represent the positive, skewed nature of nonzero rainfall; it must cover the entire range of intensities, not just the upper tail; it must be flexible enough to allow different trend magnitudes and even opposite trend directions in the bulk and the tail; and it should achieve all this with as few free parameters as possible to limit estimation uncertainty. Those conditions immediately rule out the symmetric Gaussian, the tail-only extreme value distributions, and overly complex models with too many parameters.

Three candidate distributions emerged. The two-parameter gamma distribution is the most popular choice in the hydro-climatological literature, prized for its simplicity, though previous global surveys have shown its tail is often too light to capture the heaviest observed rainfall. The three-parameter generalized gamma adds a shape parameter that controls tail heaviness, allowing heavy-tailed, light-tailed, or bounded behavior, and was recommended as a primary choice for daily precipitation in a landmark analysis of more than 15,000 datasets worldwide. Finally, the extended generalized Pareto distribution extends the classical tail model to the full range of nonzero intensities while remaining consistent with extreme value theory in both tails, with separate parameters governing the lower tail, the spread, and the upper tail.

The real innovation lies in how the team made these distributions nonstationary, meaning their parameters evolve over time. Rather than assuming rainfall statistics are fixed, they allowed the distribution parameters to respond linearly to sea surface temperature anomalies, drawn from the NOAA Extended Reconstructed Sea Surface Temperature dataset and averaged over the Mediterranean and nearby Atlantic. Warm sea surfaces increase evaporation and moisten the atmosphere, providing a physically meaningful proxy for the thermodynamic changes accompanying global warming. Crucially, the researchers tested variants in which only the location or scale parameters varied, and variants in which the shape parameter, the component controlling tail behavior, also evolved. Standard practice in extreme value analysis usually keeps the shape parameter fixed out of concern for estimation stability, but because this framework uses every nonzero daily observation rather than only annual maxima, the data volume supports the extra flexibility.

The testing ground was metropolitan France, an ideal natural laboratory spanning oceanic, continental, mountainous, and Mediterranean climates. The team analyzed autumn precipitation, the season when the country’s most intense extremes occur, at 934 rain gauges operated by Météo-France and Électricité de France, with record lengths of 64 to 73 years spanning 1950 to 2022. The EDF stations, concentrated in the Alps, Pyrenees, and Massif Central for dam management purposes, had been carefully homogenized using a combination of statistical homogeneity tests. Model selection proceeded in two stages: first the Akaike Information Criterion, which balances goodness of fit against model complexity, was used to identify the best variant of each distribution family, and then the finalists were judged on their ability to reproduce trends across low, medium, and extreme quantiles.

The results were unambiguous. The generalized gamma won the Akaike comparison at 68 percent of stations, with the extended generalized Pareto preferred at 22 percent and the plain gamma at just 10 percent. More revealing were the trend-reproduction diagnostics. Comparing model outputs against independent nonparametric benchmarks, quantile regression for the quantile trends, the Theil-Sen slope estimator for mean trends, and a nonstationary generalized extreme value model for ten-year return levels, the team found that only the most flexible variants, those with an evolving shape parameter, could capture the full richness of observed change. At some stations, low quantiles are falling while high quantiles rise; at others, such as Paris-Montsouris, trends flip direction twice across the quantile range. Simple models forced a single trend onto every quantile, while the flexible generalized gamma and extended generalized Pareto variants reproduced even these tri-directional patterns, achieving concordance correlation coefficients of around 0.8 against the extreme value benchmark for return-level trends, compared with roughly 0.4 to 0.55 for less flexible alternatives.

The resulting map of French autumn rainfall is a mosaic of change. Wet-day frequency has increased across most of the country, and mean all-day precipitation shows significant increases nearly everywhere except the southwest. Extreme precipitation tells a more worrying regional story: significant positive trends in the ten-year return level appear along the Rhône valley, in the southeast, and around Brittany, while significant negative trends emerge in the northern Alps and the western Massif Central. These spatially heterogeneous signals, the authors argue, point to geographically distinct climate drivers and reinforce the case for localized, nonstationary modeling rather than one uniform model imposed across diverse climatic zones.

Beyond the immediate findings, the study carries a broader message for how climate impact analysis should be done. Because a single flexible distribution describes the entire precipitation spectrum, the framework avoids the artificial separation between bulk and tail that plagues two-component models, and because its parameters are conditioned on covariates like sea surface temperature, it can be forced with climate model projections to generate physically consistent synthetic rainfall for the future. The researchers plan to extend the analysis with a deeper investigation of covariate selection and regional drivers. For now, the takeaway for the hydrology community is clear: if you want to know how both the drizzle and the deluge are changing, let the shape of the distribution change with it.

Subject of Research: Nonstationary statistical modeling of trends across the full spectrum of daily precipitation under climate change

Article Title: Selecting the best distribution for modeling trends in low, medium, and extreme daily precipitation under climate change

Article References: Haruna, A., Blanchet, J., Evin, G., & Paquet, E. (2026). Selecting the best distribution for modeling trends in low, medium, and extreme daily precipitation under climate change. Advances in Statistical Climatology, Meteorology and Oceanography, 12(1), 87-109. https://doi.org/10.5194/ascmo-12-87-2026

Image Credits: AI Generated

DOI: 10.5194/ascmo-12-87-2026

Keywords: precipitation, climate change, nonstationary modeling, extreme value theory, generalized gamma distribution, extended generalized Pareto distribution, Akaike Information Criterion, sea surface temperature, France, flood risk, hydrology, quantile regression