Changes in precipitation patterns, particularly in the frequency and intensity of both mean and extreme events, represent one of the most consequential aspects of anthropogenic climate change . Detecting and modeling such changes are critical for improving hydrological forecasting, assessing flood and drought risks, and informing infrastructure planning and climate adaptation strategies.

Accurately assessing these trends requires robust statistical models that can capture the complex behavior of daily precipitation. Daily precipitation poses well-known statistical challenges. It is non-negative, skewed, exhibits a discrete-continuous nature with periods of no rainfall (dry-days) interspersed with varying intensities of rainfall (wet-day), and often shows heavy tails . Traditional statistical approaches often simplify this complexity, either by focusing solely on trends in mean precipitation e.g., or by treating only extreme events using nonparametric approaches e.g. or parametric techniques based on Generalized Extreme Value (GEV) distribution e.g., or generalized Pareto (GPD) e.g.. However, in applications such as stochastic simulation, where the marginal distribution of all the precipitation amounts is required, it is necessary to model the trends in the entire precipitation distribution. Additionally, employing the entire precipitation data enables trend assessment within a single statistical framework. This ensures coherence by explicitly representing trends for all quantiles and provides robustness by leveraging the full long time series of daily observations. This constitutes an advantage over methods like the Theil-Sen slope estimator and the Mann-Kendall test , which separately estimate trends in mean (using only time series of means) and extremes (using only annual maxima series).

Modeling the full spectrum of daily precipitation, accounting for its intermittency, can be achieved using a mixed-type distribution , comprising a discrete component for dry-days and a continuous component for wet-day precipitation. The discrete component is represented by a probability mass concentrated at zero, while the continuous component is described by a parametric distribution function. Representing the marginal distribution of wet-day precipitation by a parametric distribution is essential for making extrapolations beyond the recorded values, for stochastic simulations in rainfall generators, or nonstationarity analysis. Conversely, an empirical function, based on plotting position formulas see, would suffice if the interest lies solely in the estimation of stationary quantiles within the bulk of the observed distribution, far from its tail.

When modeling trends in whole daily precipitation distribution, choosing the right marginal distribution is crucial not only for fitting the data but also for accurately capturing its temporal evolution. An ideal distribution for this task must meet several conditions: (i) it needs to represent the positive and skewed nature of nonzero daily precipitation; (ii) it should be capable of modeling the entire range of nonzero precipitation, not just the upper tail; (iii) it needs to be flexible enough to handle different trend magnitudes and directions across the bulk and tail; and (iv) finally, it should achieve this with the fewest possible free parameters to minimize estimation uncertainties. The first condition excludes the commonly used Gaussian distribution due to its symmetry. The second condition rules out the classical extreme value theory distributions, such as the GEV and GPD, which are tailored to the tail of the distribution. The third condition excludes one-parameter distributions, such as the exponential distribution, because they lack the flexibility to model multiple trend directions in the distribution. The last condition finally excludes distributions with many parameters, such as kappa, leading to increased complexity and estimation uncertainty.

Several parametric distributions could potentially satisfy these four conditions; however, considering them all would be impractical and computationally prohibitive. For this study, we consider three prominent candidate models with two or three parameters widely applied in hydro-climatological literature: gamma, generalized gamma , and the extended generalized Pareto distribution . An additional advantage of selecting these distributions is their hierarchical nature, as many other commonly used distributions are special cases. For instance, the exponential distribution is a special case of both the gamma and the extended generalized Pareto, while, depending on the parametrization, Weibull, gamma, exponential, and lognormal distributions can all be derived as special cases of the generalized gamma.

The gamma (GA) distribution is a widely adopted choice for modeling precipitation due to its simplicity and flexibility, recognized as the most popular option in the literature . As a two-parameter model belonging to the exponential family, its upper tail's behavior (exponential, slightly lighter, or heavier) depends on its shape parameter. However, numerous studies have shown that the upper tail of observed precipitation often exhibits heavier characteristics than GA model can adequately capture , leading to an underestimation of the magnitude and frequency of heavy precipitation events . Nevertheless, GA remains a cornerstone within the hydro-climatological community, finding extensive applications in stochastic modeling e.g., trend analysis e.g., and frequency analysis e.g..

The generalized gamma (GG) distribution is a three-parameter model that has also seen broad application in precipitation modeling e.g.. A comprehensive global analysis of over 15 000 daily precipitation datasets by found the GG distribution to flexibly model nonzero precipitation, leading the authors to recommend it as a primary choice for daily precipitation modeling. Unlike the two-parameter GA, the GG offers enhanced flexibility, capable of modeling heavy-tailed, light-tailed, or bounded distributions depending on the value of its shape parameter.

The extended generalized Pareto distribution (EGPD) is a family of models that extends the classical GPD, which is typically tailored for extremes above a high threshold, to model the entire range of nonzero amounts. This extension ensures the EGPD remains consistent with extreme value theory in its upper and lower tails while providing full-range applicability. Among the EGPD family, we consider the three-parameter model based on a power law that has been widely applied across various fields. Its utility has been demonstrated in modeling daily precipitation see, sub-daily precipitation e.g., supra-daily precipitation e.g., sub-hourly precipitation e.g., and even in diverse applications such as wave height modeling and wildfire analysis using deep graphical regression e.g.. Furthermore, it has been successfully integrated into nonstationary frameworks e.g.. More recently, proposed a zero-inflated EGPD model that unifies the modeling of dry days, low, moderate, and extreme rainfall within a single framework.

The objective of this study, therefore, is to identify which of these candidate distributions most adequately models the observed trends across the entire range of daily precipitation, from its mean to its extremes. While comparisons of distributions for modeling daily precipitation have been conducted at both regional e.g. and global scales e.g., these efforts have generally been confined to a stationary framework. Our study extends this crucial research by performing such comparisons within a nonstationary framework, explicitly addressing the temporal evolution of precipitation characteristics. Furthermore, our model selection methodology transcends typical goodness-of-fit criteria, incorporating a multi-criterion assessment of each model's flexibility in accurately capturing trends across low, medium, and extreme precipitation quantiles.

We apply the framework to a dense network of meteorological stations in France. This region provides a suitable candidate given the wide variety of climatic regimes, ranging from mountainous, oceanic, continental, and Mediterranean. In addition, it also has a dense network of more than 900 stations, with an average of 70 years of data, enabling the conclusions drawn from this study to be considered broadly generic. The remainder of this article is organized as follows: Sect.  presents the data and the study area. The methodology, model inference, selection, as well as uncertainty analysis, are presented in Sect. . The results are presented in Sect.  while discussion are done in Sect. . Finally, the conclusions and some relevant perspectives are given in Sect. .

We have a daily precipitation dataset from a total of 934 rain gauges spread across metropolitan France, shown in Fig. . The data is sourced from the Meteo-France networks and those of Électricité de France (EDF). From the Figure, the EDF stations are primarily situated in mountainous regions, including the Alps, Pyrenees, and Massif Central, for addressing the need of dam management. The data has been screened and homogenized using an EDF-developed tool that combines Alexanderson's homogeneity test , Bois Ellipse , and linear regression methods . The length of the series ranges from 64 to 73 years from the period 1950 to 2022. In this study, we consider only data coming from the autumn season, which we define as consisting of October, November, and December. Autumn is known to be the season when heavy precipitation is observed, particularly in the southern region where the most intense extremes occur. To distinguish between dry and wet-day, we use a threshold of 1 mm to reduce the uncertainty in the recording procedure. This threshold of 1 mm is commonly used in the literature e.g..

To explain the observed trends in wet-day precipitation, we incorporate Sea Surface Temperature (SST) anomaly as a covariate within our nonstationary models. Warm SST increases the turbulent near-surface heat fluxes that moisten and destabilize the atmosphere, thereby increasing evaporation and the intensity of convection and precipitation amounts . SST has been widely used as a covariate to explain changes in precipitation e.g. and in the south of France by . While other covariates have been used in the literature e.g. climatic covariates in , we consider their specific selection to be largely irrelevant for the comparative analysis presented in this study, as all models use the same SST series. In any case, SST provides a stable representation of the thermodynamic changes associated with global warming, allowing for a focused comparison of the internal flexibility of the candidate distributions without the confounding effects of model-selection uncertainty across multiple predictors. A more detailed analysis of covariate selection, including potential regional variations and alternative climate drivers, will form the subject of a future communication.

The SST data we employ in the study was obtained from the NOAA-NCDC Extended Reconstructed Sea Surface Temperature Dataset Version 5 . We spatially averaged the SST anomalies over the Mediterranean and the proximate Atlantic, specifically spanning longitude 28° W to 19° E and latitude 36 to 58° N (see Fig. ) . Given our focus on long-term trends, the annual SST values were smoothed using nonparametric locally weighted scatter plot smoothing (LOWESS), implemented via the lowess function in the . Figure  illustrates both the raw and smoothed annual SST and the window over which we take the spatial average.

This section presents the statistical models that we use, the inference procedure, the comparison framework, as well as the method for uncertainty evaluations.

This study presents a comprehensive comparison framework for selecting the best nonstationary model of daily precipitation using a mixed discrete-continuous distribution, driven by SST and time as a key covariates. By evaluating various distributional families (GA, GG, EGPD) and their nonstationary variants, our approach moves beyond traditional stationary or simpler nonstationary methods to provide a more refined understanding of precipitation trends. We have chosen France as a case study due to its wide diversity of climates with different influences (Atlantic in the West, Mediterranean in the South, continental in the East) and a complex topography (Pyrenees, Massif Central, Alps). It also presents a dense network of long time series of observed precipitation, which enables the regional characterization of the trends. The multi-criteria model selection process, combining AIC with rigorous diagnostic comparisons of trend evolutions, has revealed critical insights into the best-performing models and the nature of precipitation nonstationarity across France.

A central finding of our AIC-based model selection is the evidence for nonstationarity in autumnal french daily precipitation (either in the dry-day frequency, wet-day distribution or both), with the stationary case being rarely selected (Fig. ). This shows the inadequacy of the stationary model and the need to incorporate evolving hydro-climatic conditions in climate impact studies and assessments. Furthermore, our intramodel comparisons revealed a strong spatial coherence in the form of nonstationarity, irrespective of the wet-day distribution assumed. The prevalence of trends in dry-day frequency (in isolation or alongside wet-day intensity trends) in northern and western France, contrasting with wet-day intensity-only trends in the southwest, suggests the possibility of geographically distinct climate drivers. This regional heterogeneity reinforces the need for localized nonstationary modeling rather than applying a single, uniform model across diverse climatic zones. For a regional study where continuous parameter maps are required, the most flexible model that accounts for all scenarios of nonstationarity, might be more appropriate.

The intermodel comparison conclusively demonstrated the superior performance of the GG distribution, which was selected by AIC at 68 % of stations. This widespread preference for GG can be attributed to its optimal balance of flexibility and parsimony. As a three-parameter distribution, GG offers greater adaptability to capture a wider range of shapes (e.g., varying skewness and kurtosis, inclusive of heavy tails) inherent in precipitation data compared to the two-parameter GA distribution. The absence of a clear spatial pattern for GA selections implies that where it is preferred, it likely reflects highly localized data characteristics that do not consistently align with broader geographical or climatic influences, rather than a systematic regional fit. This finding aligns with growing evidence in hydrological modeling that more flexible distributions, like the GG, are often better suited for complex environmental data e.g.,, especially precipitation with its high skewness.

While EGPD is also a three-parameter model designed for robust tail modeling (with κ controlling the lower tail, σ the spread, and ξ the upper tail), its lower overall selection rate by AIC compared to GG, despite similar flexibility, suggests that the GG's family of shapes may better represent the overall characteristics of daily precipitation in most French regions. This is supported by Fig.  where the EGPD showed slightly lower performance compared to GG in the lower tails. However, the concentrated selection of EGPD in specific areas like the northeast of the Massif Central indicates that its unique tail-capturing capabilities, particularly through its ξ parameter, are indeed necessary for regions with distinct extreme precipitation characteristics. Figure  highlight that precipitation in these regions is heavy-tailed, as measured by the shape parameters of stationary GG, EGPD, and GEV.

Beyond AIC, our diagnostic evaluations of trend reproduction further corroborated these findings. The ability of the most flexible GG-Mpd,μ,σ,ν and EGPD-Mpd,σ,κ,ξ variants to accurately reproduce the complex, often non-monotonic, trends in mean, quantile, and return level evolutions, especially in the tails, provides compelling empirical support for their increased flexibility. This was particularly evident in the tri-directional quantile trends observed at stations like Paris-Montsouris (Fig. ), which only the most flexible models could capture, while less flexible or simpler distributions like GA often exhibited significant biases. The strong agreement of these models with nonparametric QR and nonstationary GEV benchmarks underscores their robustness. This comprehensive validation addresses a crucial gap in many nonstationary studies that rely solely on information criteria for model selection.

It is pertinent to discuss our findings on parameter nonstationarity in the context of traditional extreme value theory applications. In many studies employing the GEV or GPD for extreme precipitation analysis, trends are often modeled by allowing nonstationarity in only the location (μ) or scale (σ) parameters, or both e.g.,. The shape parameter (ξ) is frequently kept stationary, partly due to concerns about its estimation stability given limited extreme data points. However, in our mixed-distribution framework, the entire wet-day distribution is modeled, implying that the location and scale parameters (e.g., of the GG or EGPD) are heavily influenced by the bulk of the distribution, not just the extremes. Consequently, to adequately capture complex trends specifically in the extremes of the distribution, allowing for nonstationarity in the shape parameter (e.g., ν for GG, and particularly ξ for EGPD) becomes crucial. Our results, particularly the superior performance of the most flexible models GG-Mpd,μ,σ,ν and EGPD-Mpd,σ,κ,ξ in accurately reproducing trends in the upper quantiles and return levels, empirically support the necessity of this additional flexibility in the shape parameter to adequately characterize nonstationary in the upper tail of the precipitation distribution.

The observed trends with the best-performing GG-Mpd,μ,σ,ν model reveal significant and spatially heterogeneous changes in daily precipitation characteristics across France. The widespread significant increase in wet-day frequency and mean all-day precipitation across most of the study area, coupled with a notable increase in wet-day intensity in the southeast, points towards a general moistening trend. However, the significant positive trends in the 10-year return levels in regions like the Rhône Valley and the southeast are particularly concerning. These findings indicate an intensification of extreme precipitation events, which carries substantial implications for flood risk management, agricultural planning, and water resource infrastructure in these vulnerable regions.

Our study aimed to identify a suitable distribution that can flexibly model the observed trends in the entire range of daily precipitation distribution. Given the discrete-continuous nature of daily precipitation, resulting from the occurrence of dry and wet-day, we employ a mixed distribution of discrete type for the dry component, and continuous type for the nonzero component. We identify three distributions commonly used in the literature for modeling nonzero precipitation: gamma, generalized gamma, and extended generalized Pareto (EGPD). Using these distributions, we form a number of nonstationary models, accounting for three main forms of nonstationarity: (i) Trend in only the dry-day frequency, (ii) Trend in only the wet-day distribution, and (iii) Trend in both the dry-day frequency and the wet-day distribution. Irrespective of the given distribution, we employ time as a covariate for the dry-day component, and sea surface temperature for parameters of the wet-day precipitation distribution.

We employ an approach based on information theory, utilizing AIC to select the best nonstationary model, complemented by an assessment of each model's capacity to flexibly capture trends across low, medium, and extreme quantiles of the precipitation distribution. As a case study, we consider autumnal daily precipitation in France at over 900 stations. The main findings of the study can be summaried below:

Irrespective of the chosen distribution, the resulting form of nonstationarity identified by AIC was regionally coherent, with the models involving a trend in the dry-day frequency selected in most of the study area.

These findings underscore the importance detailed and multi-criterion approach to model identification in trend analysis, particularly under changing climatic conditions, due to the resulting implications for both hydrological and climate impact assessments. Our study contributes to the critical challenge of flexibly modeling observed trends across the entire distribution of daily precipitation, a task complicated by its inherent discrete-continuous nature. With regards to stochastic weather generators, a major challenge is the seamless representation of both frequent low-intensity events and rare extremes. By using a single flexible distribution (like the EGPD or GG) that maintains statistical consistency across the full spectrum, these models avoid the artificial discontinuities often found in “split-model” generators (e.g., Gamma for the bulk and GPD for the tail). Because our model parameters are conditioned on covariates, these models are uniquely capable of generating non-stationary synthetic series. By forcing the models with GCM-derived SST or other relevant covariates projections, they can produce future stochastic rainfall realizations that account for thermodynamically and dynamically driven shifts in both mean and extreme intensities. Building on these results, the next phase of our research will involve a detailed trend analysis of French daily precipitation using the generalized gamma model. This will include comprehensive covariate selection, involving additional large-scale climate indices or local drivers, and evaluation of regionally varying influences, ultimately contributing to more robust climate adaptation strategies for critical sectors such as water resource management, agriculture, and flood risk assessment.