Zheng et al. (2026) Rainfall frequency analysis: interval estimations for Weibull distribution based on specially formulated pivotal quantities

Identification

• Journal: Theoretical and Applied Climatology
• Year: 2026
• Date: 2026-01-01
• Authors: Weiqiang Zheng, Shuguang Liu, Zhengzheng Zhou
• DOI: 10.1007/s00704-025-05916-y

Research Groups

• College of Civil Engineering, Tongji University, Shanghai, China
• State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China
• Key Laboratory of Cities’ Mitigation and Adaptation to Climate Change in Shanghai, China Meteorological Administration, Shanghai, China
• Shanghai Key Laboratory of Urban Regeneration and Spatial Optimization Technology, Tongji University, Shanghai, China

Short Summary

This study introduces a novel Pivotal Quantity Method (PQM) using three specially formulated joint pivotal quantities for the three-parameter Weibull distribution to improve confidence interval estimations for parameters and return levels in rainfall frequency analysis. Through simulations and application to rainfall data in Jiangsu Province, China, the PQM is shown to provide more reliable and robust interval estimations, especially for small sample sizes, compared to conventional bootstrap methods.

Objective

• To design and evaluate three specially formulated joint pivotal quantities for the three-parameter Weibull distribution to obtain more reliable confidence interval (CI) estimations for its parameters (location ξ, scale β, shape c) and return levels.
• To compare the performance of the proposed Pivotal Quantity Method (PQM) against standard nonparametric and parametric bootstrap methods, particularly under conditions of small sample sizes.

Study Configuration

• Spatial Scale: 11 rain gauges in Jiangsu Province, China.
• Temporal Scale: 56-year (1965–2020) annual maximum series (AMS) for the application; simulation experiments with sample sizes (n) of 20, 40, 60, and 100.

Methodology and Data

• Models used:
  • Three-parameter Weibull distribution for rainfall frequency analysis.
  • Pivotal Quantity Method (PQM) based on three newly formulated joint pivotal quantities.
  • Nonparametric Bootstrap Method (NBM) and Parametric Bootstrap Method (PBM) for comparison.
  • L-moment method for point estimation in bootstrap methods and fitting the Weibull distribution in PBM.
  • Kolmogorov–Smirnov (KS) test for goodness-of-fit assessment.
• Data sources:
  • Synthetic data generated from known parent Weibull distributions for simulation experiments.
  • Long-term daily rainfall dataset (1965–2020) from the China Meteorological Data Service Center (http://data.cma.cn/) for 11 rain gauges in Jiangsu Province, China. Annual Maximum Series (AMS) were extracted from this dataset.

Main Results

• Parameter Interval Estimations: PQM consistently demonstrates superior reliability for confidence interval estimations of Weibull distribution parameters (ξ, β, c) when the sample size (n) is small. For larger n, NBM and PBM show advantages for ξ and β, respectively.
• Return Level Interval Estimations: PQM performs better than NBM and PBM in providing reliable interval estimations for return levels, particularly when either the sample size (n) or the significance level (α) is small. PQM remains competitive even with large n and small α.
• Application in Jiangsu Province: PQM exhibits high efficiency in estimating intervals for the location parameter (ξ) and yields reliable point estimations for design rainfall, with its mean/median realizations closely aligning with L-moment-based results.
• Robustness: PQM provides more robust and stable interval estimates for design rainfall, especially in cases where conventional point estimation methods tend to underestimate or when extreme outliers are present in the data.
• Regional Characteristics: A larger shape parameter (c) is associated with wider interval estimations for the location parameter (ξ), while a smaller coefficient of variation (Cv) correlates with wider interval estimations for the shape parameter (c).

Contributions

• Introduces a novel approach for confidence interval estimation in rainfall frequency analysis by formulating three joint pivotal quantities for the three-parameter Weibull distribution.
• Develops the Pivotal Quantity Method (PQM) that effectively circumvents the need for large-sample conditions, prior distributions, or likelihood ratio tests, which are often required by conventional CI estimation methods.
• Demonstrates, through extensive simulation experiments and real-world application, that PQM provides more reliable and robust confidence interval estimations for both parameters and return levels, particularly under challenging small-sample conditions.
• Offers a method that can generate rainfall realizations and interval estimates without relying on specific point estimators, thereby reducing the influence of point estimation procedure choices on the results.
• Highlights the potential for extending the proposed pivotal quantities to other generalized distributions commonly used in hydrological statistics, such as the Generalized Extreme Value (GEV) distribution.
• Provides a valuable complementary tool for hydrological engineers to determine more reliable design rainfall estimates, thereby enhancing safety margins in hydraulic and flood-control design.

Funding

• National Natural Science Foundation of China (Project codes: 42271031 and 42371030)

Citation

@article{Zheng2026Rainfall,
  author = {Zheng, Weiqiang and Liu, Shuguang and Zhou, Zhengzheng},
  title = {Rainfall frequency analysis: interval estimations for Weibull distribution based on specially formulated pivotal quantities},
  journal = {Theoretical and Applied Climatology},
  year = {2026},
  doi = {10.1007/s00704-025-05916-y},
  url = {https://doi.org/10.1007/s00704-025-05916-y}
}