The mathematical foundation of the study lies in extreme value theory, a branch of statistics developed specifically to model the behavior of rare events rather than averages. The Generalized Extreme Value, or GEV, distribution works by taking the single largest daily rainfall value from each year and fitting a flexible probability curve to those annual maxima. That curve has three variants named after their discoverers: the Gumbel family, with a moderately heavy tail; the Fréchet family, whose unbounded tail allows for extraordinarily large extremes; and the Weibull family, whose tail is bounded, meaning there is a physical ceiling on how extreme an event can become. Which variant applies in a given region is governed by the shape parameter, and estimating it correctly is the crux of any return-period calculation, because the tail behavior determines how quickly probabilities decay as you project toward rarer and rarer storms.

The Peak Over Threshold, or POT, approach takes a fundamentally different view of the data. Instead of discarding all but one value per year, it captures every daily rainfall that exceeds a chosen high threshold, thereby using far more information about the upper tail of the distribution. The exceedances are then modeled with the Generalized Pareto distribution, a technique rooted in Picklands’ foundational 1975 work on extreme order statistics. The trade-off is well known among hydrologists: POT extracts more signal from limited records and typically yields narrower uncertainty intervals, but its results are sensitive to how the threshold is selected, whereas GEV discards potentially useful secondary storms within a single year yet is simpler and more standardized in practice.

To capture India’s enormous climatic diversity, the authors divided the country into six homogeneous regions: north, northeast, west, east, central, and peninsular India. This regional partition matters because the drivers of extreme rainfall differ dramatically across the subcontinent, from the orographic uplift along the Western Ghats and the Himalayan foothills to the depression-driven systems that sweep across the Gangetic plains. The rainfall data came from the gridded daily product maintained by the India Meteorological Department, providing a spatially consistent record spanning nine decades, a period during which global warming, rapid urbanization, and land-use change have all reshaped the hydrological cycle.

The shape parameter results reveal a striking geographic split. In Northeast India, one of the wettest places on the planet, the fitted GEV shape parameter was approximately zero, which corresponds to an effectively Gumbel-type tail: extremes grow with rainfall intensity but without the explosive amplification of a heavy-tailed Fréchet distribution. In the other five regions, the shape parameter was negative, indicating bounded Weibull-type tails in which the largest conceivable daily rainfall approaches a finite upper limit. For flood engineers, this distinction is far from academic. A bounded tail implies that 100-year and 1000-year return levels converge toward a ceiling, while a heavier tail keeps pushing design storms upward as the return period lengthens, with direct consequences for the safety margins built into critical infrastructure.

The estimated 100-year return levels quantify just how uneven India’s extreme rainfall landscape is. In western India, the GEV-based 100-year daily rainfall came out at roughly 358 millimeters, with a 95 percent bootstrap confidence interval spanning 308 to 409 millimeters. In northeastern India, the corresponding figure soared to about 1107 millimeters, with an interval of 923 to 1353 millimeters, meaning that in the worst plausible case a single day’s rain in the northeast could exceed the entire annual rainfall of many semi-arid regions. These numbers translate directly into design guidance: a culvert sized for western Indian extremes would be dangerously undersized if transplanted to the Brahmaputra valley, and the wide confidence intervals in the northeast underscore how much statistical uncertainty remains even with ninety years of data.

When the two modeling philosophies were compared directly, the POT method emerged as the more accurate fit in every one of the six regions. Its 100-year return level estimates ranged from about 357 millimeters to 1045 millimeters, closely tracking the GEV values but generally with tighter precision, a benefit attributable to its richer use of threshold exceedances. This finding aligns with a growing body of international literature suggesting that threshold-based models, when their thresholds are chosen carefully, offer the best of both worlds: the theoretical rigor of extreme value theory combined with the statistical efficiency of using multiple extreme events per year rather than just the annual maximum.

Perhaps the most methodologically important contribution of the study is its insistence on out-of-sample validation. The authors split the record into a training period from 1930 to 1999 and a testing period from 2000 to 2019, then evaluated how well models calibrated on the historical data predicted the extremes of the most recent two decades, using the negative log-likelihood as the scoring metric. The results exposed regional differences in predictive calibration that would have remained invisible under conventional in-sample goodness-of-fit testing. A model can fit the data it was trained on beautifully and still misjudge the tail behavior of the future, and the study demonstrates that out-of-sample evaluation is a genuinely better guide to real-world flood risk than fit statistics computed on the same data used for estimation.

The practical stakes of this work are considerable. Floods rank among the most frequent and destructive natural hazards worldwide, threatening lives, ecosystems, infrastructure, and economic stability, and their severity is being amplified by climate change, urbanization, and hydrological variability. Previous research, including landmark analyses of increasing extreme rain events over India in a warming environment, has documented that the character of Indian rainfall extremes has been shifting, which makes the choice of statistical model even more consequential: an outdated or poorly validated tail model can silently understate the risk that a dam, levee, or urban drainage network will be overwhelmed. By providing a regionally consistent assessment that explicitly separates model fit from predictive skill, the study gives Indian water managers a clearer map of where each tool can be trusted.

The research also opens doors for future refinement. The analysis assumes stationarity, treating the 1930 to 2019 record as drawn from an unchanging climate, whereas a growing literature on non-stationary extreme rainfall over Indian cities suggests that allowing distribution parameters to evolve with time or with climate covariates could sharpen return-level estimates further. Extending the out-of-sample framework to non-stationary models, to finer spatial resolutions, and to sub-seasonal extremes would be natural next steps. For now, the message of this study is clear and actionable: for diagnosing extreme precipitation across India, the Peak Over Threshold method offers the most accurate regional fits, the northeast demands special caution because of its extraordinary rainfall magnitudes and wide uncertainty bands, and no flood risk assessment should rely on in-sample fit alone when the true test is how well a model anticipates the storms of the next twenty years.

Subject of Research: Extreme value analysis of extreme precipitation and return levels across India using GEV and POT methods

Article Title: Assessing the suitability of GEV and POT for extreme precipitation diagnostics over India

Article References: Jain, J., Muñoz-Arriola, F., & Khare, D. (2026). Assessing the suitability of GEV and POT for extreme precipitation diagnostics over India. Theoretical and Applied Climatology, 157(11), Article 693. https://doi.org/10.1007/s00704-026-06643-8

Image Credits: AI Generated

DOI: 10.1007/s00704-026-06643-8

Keywords: extreme precipitation, extreme value theory, Generalized Extreme Value distribution, Peak Over Threshold, return levels, flood risk, India, monsoon, Generalized Pareto distribution, out-of-sample validation, hydroclimatology, Theoretical and Applied Climatology