In many random phenomena, such as life-testing experiments and environmental data (like rainfall data), there are often positive values and an excess of zeros, which create modeling challenges. In life testing, immediate failures result in zero lifetimes, often due to defects or poor quality, especially in electronics and clinical trials. These failures, called zero inliers, are difficult to model using standard approaches. When studying extreme values in the above scenarios, a key issue is selecting an appropriate threshold for accurate tail approximation of the population using asymptotic models. While some extreme value mixture models address threshold estimation and tail approximation, conventional parametric and non-parametric bulk and generalised Pareto distribution (GPD) approaches often neglect inliers, leading to suboptimal results. This paper introduces a framework for modeling extreme events and inliers using the GPD, addressing threshold uncertainty and effectively capturing inliers at zero. The model's parameters are estimated using the maximum likelihood estimation (MLE) method, ensuring optimal precision. Through simulation studies and real-world applications, we demonstrate that the proposed model significantly outperforms the traditional methods, which typically neglect inliers at the origin.

翻译：在许多随机现象中，如寿命测试实验和环境数据（如降雨量数据），常存在正值和大量零值，这给建模带来了挑战。在寿命测试中，即时失效导致寿命为零，这通常源于缺陷或质量低劣，尤其在电子产品和临床试验中。这些被称为零内点的失效难以用标准方法建模。在上述场景中研究极值时，一个关键问题是选择合适的阈值，以便使用渐近模型对总体尾部进行准确逼近。虽然一些极值混合模型处理了阈值估计和尾部逼近问题，但传统的参数化和非参数化主体分布与广义帕累托分布方法常忽略内点，导致结果欠佳。本文提出了一种使用广义帕累托分布对极端事件和内点进行建模的框架，解决了阈值不确定性并有效捕捉零值处的内点。模型参数采用最大似然估计方法进行估计，确保了最优精度。通过模拟研究和实际应用，我们证明所提出的模型显著优于传统方法，后者通常忽略原点处的内点。