Modeling Extreme Events in the Presence of Inlier: A Mixture Approach

TL;DR

The paper tackles extreme-value analysis when inliers occur at zero by introducing the Extreme Value Inlier Mixture Model (EVIMM), which combines a degenerate mass at zero, a gamma bulk below a threshold $u$, and a generalized Pareto tail above $u$. Threshold uncertainty is incorporated by treating $u$ as an estimable parameter within a maximum-likelihood framework. The authors provide an explicit data-generation algorithm, derive the MLEs, and validate the approach through extensive simulations and real-data applications, showing EVIMM reduces bias and MSE and improves return-level estimates relative to EVMM, which neglects inliers. The work offers practical gains for reliability, environmental, and risk-management data where zero-inliers and extremes co-occur, and suggests future extensions to broader inlier patterns and Bayesian estimation.

Abstract

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.

Figures (7)

• Figure 1: Distribution of an EVMM with bulk density $h$ and a GPD as tail density.
• Figure 2: Distribution of the EVIMM for various parameter values: $\alpha \in \{0, 0.1, 0.2, 0.3, 0.4\}$, $\eta = 1$, $\beta = 5$, $\xi \in \{0.2, 0, -0.2\}$, $\sigma = 5$, and threshold $u$ corresponding to the 83rd to 90th quantiles of a gamma distribution, such that the GPD proportion is 0.1.
• Figure 3: Tail behaviour of the EVIMM for various parameter values, with a zoomed-in view of the tail region. Parameters: $\alpha \in \{0, 0.4\}$, $\eta = 1$, $\beta = 5$, $\xi \in \{0.2, 0, -0.2\}$, $\sigma = 5$, and threshold $u$ corresponding to the 83rd to 90th quantiles of a gamma distribution, ensuring a GPD proportion of 0.1.
• Figure 4: Bias comparison plot between the new model (extreme value inlier mixture model, EVIMM) and the literature model (extreme value mixture model, EVMM).
• Figure 5: MSE comparison plot between the new model (extreme value inlier mixture model, EVIMM) and the literature model (extreme value mixture model, EVMM).