[Paper Review] A parsimonious tail compliant multiscale statistical model for aggregated rainfall

Pierre Ailliot, Carlo Gaetan+1 more|arXiv (Cornell University)|Jan 13, 2026

Hydrology and Drought Analysis0 citations

TL;DR

The paper develops a parsimonious multiscale rainfall model based on extended generalized Pareto distributions (EGPD) and compound Poisson sums, enabling consistent IDF-like curves across aggregation scales and avoiding return-level crossings.

ABSTRACT

Modeling rainfall intensity distributions across aggregation scales (from sub-hourly to weekly) is essential for hydrological risk analysis and IDF curves. Aggregation naturally imposes mathematical constraints: return levels must be ordered by time scale, as daily accumulations necessarily exceed sub-daily ones. From a statistical perspective, each aggregation step should ideally not require additional parameters, yet parsimonious models describing the full distribution remain scarce, as most literature focuses on seasonal block maxima. In this study, we propose a parsimonious framework to model all rainfall intensities (low to large) across scales. We utilize the Extended Generalized Pareto Distribution (EGPD), which aligns with extreme value theory for both tails while remaining flexible for the bulk of the distribution. We establish a general result on the behavior of EGPD variables under various aggregation procedures. To overcome the difficulty of direct likelihood inference, we link the EGPD class to Poisson compound sums. This allows the use of the Panjer algorithm for efficient composite likelihood evaluation. Our approach ensures that return levels do not cross across scales and enables estimation for return periods below annual or seasonal levels. We demonstrate the method using sub-hourly series from six French stations with diverse climates. Only eight parameters are needed per station to capture scales from six minutes to three days. IDF curves above and below the annual scale are provided.

Motivation & Objective

• Motivate the need for consistent distributional modeling of rainfall intensities across aggregation scales (sub-hourly to daily).
• Propose an EGPD-based framework that preserves tail behavior under aggregation while remaining parsimonious in parameters.
• Link EGPD to compound Poisson sums to enable efficient likelihood-based inference via Panjer recursion.
• Provide a practical estimation strategy and apply the method to six French rainfall stations with August-September data.

Proposed method

• Model aggregated positive rainfall at scale d as A_d ~ EGPD(σ_d, κ, ξ, λ_d) with σ_d and λ_d increasing with d and κ, ξ fixed.
• Represent aggregation via a compound Poisson-EGPD: sum of N i.i.d. EGPD(σ, κ, ξ, B) variables, leading to EGPD(σ, κ, ξ, λ) for B(u)=u.
• Derive conditions under which transformed sums T(Y) of EGPD variables remain EGPD, preserving tail shapes (κ, ξ).
• Parametrize σ_d and λ_d as log-polynomials in log d to model non-crossing return levels across scales (Proposition 3.1).
• Estimate parameters by maximizing a composite likelihood across representative aggregation scales, using Panjer recursions for efficient density evaluation.

Experimental results

Research questions

• RQ1 How do tail and bulk properties of rainfall distributions change under aggregation across scales while preserving key tail parameters?
• RQ2 Can a parsimonious EGPD-based model, with fixed κ and ξ, capture the full distribution of aggregated rainfall (not just extremes) across multiple durations?
• RQ3 How should σ_d and λ_d vary with d to ensure non-crossing return levels and realistic IDF-type curves?
• RQ4 Can compound Poisson-EGPD provide a computationally efficient inference framework via Panjer recursion for real rainfall data?
• RQ5 How well does the proposed model fit observed rainfall across different climatological settings and durations?

Key findings

• A compound Poisson-EGPD framework yields an EGPD distribution for aggregated rainfall that preserves upper-tail behavior under aggregation.
• With B(u)=u and aggregation scale d, the model maintains constant ξ and κ across scales while σ_d and λ_d increase with d.
• Parametrizing log σ_d and log λ_d as log-polynomials in log d provides non-crossing return levels and realistic IDF curves across durations.
• Application to six French stations shows ξ in [0.15,0.35] and κ in (0.2,0.45); σ_d and λ_d increase with d, matching physical interpretation of scale and event rate.
• The model fits well for aggregation scales 30 minutes to 3 days, with some underestimation of extremes at finer scales and overestimation at some larger scales for certain stations.