Model averaging with mixed criteria for estimating high quantiles of extreme values: Application to heavy rainfall

Abstract

Accurately estimating high quantiles beyond the largest observed value is crucial for risk assessment and devising effective adaptation strategies to prevent a greater disaster. The generalized extreme value distribution is widely used for this purpose, with L-moment estimation (LME) and maximum likelihood estimation (MLE) being the primary methods. However, estimating high quantiles with a small sample size becomes challenging when the upper endpoint is unbounded, or equivalently, when there are larger uncertainties involved in extrapolation. This study introduces an improved approach using a model averaging (MA) technique. The proposed method combines MLE and LME to construct candidate submodels and assign weights effectively. The properties of the proposed approach are evaluated through Monte Carlo simulations and an application to maximum daily rainfall data in Korea. In addition, theoretical properties of the MA estimator are examined, including the asymptotic variance with random weights. A surrogate model of MA estimation is also developed and applied for further analysis. Finally, a Bayesian model averaging approach is considered to reduce the estimation bias occurring in the MA methods.

Figures (8)

• Figure 1: Illustration of the selection of $\xi_k$ values using the profile likelihood of $\xi$ and the corresponding weights, based on annual maximum daily rainfall data (unit: $mm$) from Hae-nam, Korea. (Top panel): The green curve represents the profile log-likelihood function, while the ten vertical lines indicate the selected $\xi_k$ values. (Bottom panel): The 200-year return level estimates from ten submodels (vertical dotted lines) are shown along with their corresponding weights (black dots) for the proposed MA method ('like1'). The return level estimates obtained from the MLE, LME, and MA methods are also displayed for comparison.
• Figure 2: Same as Table \ref{['sim_result']} but for the bias (left) and RMSE (right).
• Figure 3: (Left) Density plots overlaid on the relative frequency histogram and (Right) a scatterplot of observations by year with horizontal lines representing the 50-year return level estimates together with shaded regions for 90% confidence intervals. The data depict the annual maximum daily rainfall (unit: $mm$) in the Hae-nam station, Korea. In both figures, the blue solid line and the red dashed line correspond to the estimates obtained using the LME and MLE methods, respectively.
• Figure 4: Sampling distributions of the 100-year return levels from nine estimation methods, displayed as density plots overlaid on the relative frequency histogram. These distributions were obtained using 1,000 bootstrap samples from the annual maximum daily rainfall data (unit: $mm$) in Hae-nam, Korea. Acronyms for the method names are provided in Table \ref{['methods_MA']}.
• Figure 5: Quantile-per-quantile plots of nine methods fitted to the annual maximum daily rainfall (unit: $mm$) data in Hae-nam, Korea.