<p>The generalized Pareto (GP), generalized extreme value (GEV), and generalized logistic (GLO) distributions are widely employed in geoscience, engineering, and the social sciences. However, conventional estimation methods, such as the <i>L</i>-moment and maximum likelihood approaches, often impose restrictions on parameter ranges or rely on asymptotic approximations that may be unreliable with small samples or heavy tails. This study develops and extends the pivotal quantity method to enable point estimation and joint confidence region construction for the location, scale, and shape parameters of these distributions. Unlike conventional techniques, the proposed approach does not require constraints on parameter ranges and demonstrates strong performance even when the shape parameter is large in magnitude and sample sizes are limited. Simulation experiments confirm its robustness compared with the widely used <i>L</i>-moment method and maximum likelihood method. Additionally, the method provides a systematic framework for deriving joint confidence regions, offering more comprehensive uncertainty quantification. The practical implementation of the approach is illustrated using a real-world rainfall dataset. The method is broadly applicable to other special cases of the four-parameter Kappa distribution, providing a flexible and reliable tool for parameter inference in extreme value analysis.</p>