Generalized maximum‐likelihood generalized extreme‐value quantile estimators for hydrologic data

TLDR

The three‑parameter generalized extreme‑value distribution is widely used to model annual floods, rainfall, wind speeds, wave heights, snow depths, and other maxima, but small‑sample maximum‑likelihood estimators are unstable, leading researchers to favor L‑moment and recent method‑of‑moments quantile estimators that show lower root‑mean‑square error for −0.25 < κ < 0.30. By imposing a Bayesian prior that restricts the shape parameter κ to a statistically and physically reasonable range, the authors develop a generalized maximum‑likelihood (GML) estimator that avoids the generation of absurd κ values. In small‑sample examples, the GML estimator outperforms both moment and L‑moment quantile estimators for −0.4 ≤ κ ≤ 0, demonstrating substantially better performance.

Abstract

The three‐parameter generalized extreme‐value (GEV) distribution has found wide application for describing annual floods, rainfall, wind speeds, wave heights, snow depths, and other maxima. Previous studies show that small‐sample maximum‐likelihood estimators (MLE) of parameters are unstable and recommend L moment estimators. More recent research shows that method of moments quantile estimators have for −0.25 &lt; κ &lt; 0.30 smaller root‐mean‐square error than L moments and MLEs. Examination of the behavior of MLEs in small samples demonstrates that absurd values of the GEV‐shape parameter κ can be generated. Use of a Bayesian prior distribution to restrict κ values to a statistically/physically reasonable range in a generalized maximum likelihood (GML) analysis eliminates this problem. In our examples the GML estimator did substantially better than moment and L moment quantile estimators for − 0.4 ≤ κ ≤ 0.