Many researchers extend well-known statistical distributions to enhance their flexibility and better adapt to real-world data. However, such extensions often increase the complexity of the analysis of maximum likelihood estimation (MLE) when using classical numerical methods. In this study, the particle swarm optimization (PSO) algorithm is employed for MLE to estimate the stress–strength reliability (R) of the q-generalized extreme value distribution under linear normalization (q-GEVL) with type II progressive censoring. Additionally, Bayesian estimation (BE) is performed for R, utilizing both informative and non-informative priors with symmetric and asymmetric loss functions. A numerical simulation study is presented to evaluate the proposed methods. Furthermore, real estate prices in two U.S. states—Hawaii and West Virginia—are analyzed to demonstrate the adaptability of the q-GEVL distribution. This distribution is also applied to forecast future real estate prices using return-level and R approaches, compared against classical time series methods. As of mid-2025, the average property value is approximately 840, 000 in Hawaii and 170, 904 in West Virginia, aligning with the predicted R and return-level (RL) estimates at T = 72(0.98611) for West Virginia and T = 17(0.94117) for Hawaii.