Many random phenomena, including life-testing and environmental data, show positive values and excess zeros, which pose 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 inliers at zero, are difficult to model using standard approaches. The presence and proportion of inliers may influence the accuracy of extreme value analysis, bias parameter estimates, or even lead to severe events or extreme effects, such as drought or crop failure. In such scenarios, a key issue in extreme value analysis is determining a suitable threshold to capture tail behaviour accurately. Although some extreme value mixture models address threshold and tail estimation, they often inadequately handle inliers, resulting in suboptimal results. Bulk model misspecification can affect the threshold, extreme value estimates, and, in particular, the tail proportion. There is no unified framework for defining extreme value mixture models, especially the tail proportion. This paper proposes a flexible model that handles extremes, inliers, and the tail proportion. Parameters are estimated using maximum likelihood estimation. Compared the proposed model estimates with the classical mean excess plot, parameter stability plot, and Pickands plot estimates. Theoretical results are established, and the proposed model outperforms traditional methods in both simulation studies and real data analysis.

翻译：许多随机现象，包括寿命测试和环境数据，呈现出正值和过量零值，这给建模带来了挑战。在寿命测试中，即时失效会导致零寿命，这通常源于缺陷或质量低劣，在电子产品和临床试验中尤为常见。这些被称为零值内点的失效，难以用标准方法进行建模。内点的存在及其比例可能影响极值分析的准确性、使参数估计产生偏差，甚至导致干旱或作物歉收等严重事件或极端效应。在此类场景中，极值分析的一个关键问题是确定合适的阈值以准确捕捉尾部行为。尽管某些极值混合模型处理了阈值和尾部估计问题，但它们通常未能充分处理内点，导致结果欠佳。主体模型的误设可能影响阈值、极值估计，尤其是尾部比例。目前尚无统一定义极值混合模型（特别是尾部比例）的框架。本文提出了一种能够同时处理极值、内点及尾部比例的灵活模型。参数采用最大似然估计法进行估计。将所提模型的估计结果与经典平均超额图、参数稳定性图及Pickands图的估计结果进行了比较。研究建立了理论结果，并通过模拟研究和实际数据分析证明，所提模型优于传统方法。