Modeling extreme events in the presence of inliers: a mixture approach

In many real-life situations, such as life testing experiments and environmental studies, including rainfall, snowfall, streamflow, and soil moisture, responses contain an excess of zero observations, which are known as inliers at zero. Standard modeling approaches fail to provide expected results if these inliers are not considered in the model. Similar challenges arise in extreme value analysis, where accurate tail estimation depends on the appropriate threshold selection. Inliers can adversely affect both threshold estimation and tail behavior modeling if not properly considered. Although some extreme value mixture models address threshold and tail estimation, they often ignore inliers or don't account thoroughly, leading to suboptimal results. There is no unified framework for defining extreme value mixture models; the misspecification of the bulk model can affect the threshold, tail estimates, and particularly the tail proportion. This paper introduces a framework for modeling extreme events, addressing threshold uncertainty, and capturing inliers at zero. The model parameters are estimated using maximum likelihood estimation, ensuring precision. We compare the proposed model with classical mean excess and parameter stability plot estimates. Theoretical results are established, and the model's utility is shown through real applications. Simulation studies and real data examples demonstrate that the proposed model significantly outperforms traditional methods, which typically neglect inliers.