A New Mixed Distribution with Properties, Applications and Hydrological Engineering Risk Analysis under the Flood Peaks of the Canadian Wheaton River

Abstract

This article introduces the mixed Gamma Lindley (GL) distribution, combining Lindley’s failure-time modeling with Gamma’s heavy-tailed flexibility via a novel mixing method. We establish its corrected mathematical foundation, proving three key theoretical results: a Modality Theorem governing density shape, Mean Residual Life convergence to a finite bound, and classification within the Gumbel maximum domain of attraction. Global identifiability ensures consistent Maximum Likelihood Estimation. Applied to Canadian Wheaton River flood peaks (n=72), GL outperforms ten competitive models. Comprehensive risk analysis using Value-at-Risk (VaR), Tail-VaR (T-VaR), Tail Variance (TV_c), Tail Mean Variance (TMV_c), Mean of Order-P (MOO-P), and Peaks-Over-Random-Threshold VaR (PORT-VaR) confirms heavy-tailed behavior (Hill index ≈ 0.18) and provides coherent, actionable engineering thresholds for flood infrastructure design under climate uncertainty. This work bridges advanced statistical theory with practical hydrological engineering, offering a robust tool for managing extreme environmental risks under climate uncertainty.