A flexible modelling of extremes in the presence of inliers

Many real-world phenomena, including life-testing and environmental data, exhibit positive observations with an excess of zeros (referred to as inliers at zero), which pose significant modelling challenges. The presence and proportion of such inliers may affect extreme value analysis (EVA), biased parameter estimates, and distorted tail estimation. A key and persistent challenge in EVA is selecting an appropriate threshold to accurately capture tail behaviour. This paper proposes a flexible modelling framework that jointly accounts for extremes, inliers, and the tail fraction, treating both the threshold and the tail fraction as parameters. The model parameters are estimated using maximum likelihood methods. The proposed approach is compared with classical graphical tools, including the mean excess plot, parameter stability plot, and Pickands plot. Theoretical properties are established, and simulation studies and real-data applications demonstrate that the proposed model provides more accurate and reliable estimation, reducing bias and mean squared error and improving tail inference in the presence of inliers.