Multiple outlier detection in samples with exponential & Pareto tails

We introduce two ratio-based robust test statistics, max-robust-sum (MRS) and sum-robust-sum (SRS), which compare the largest suspected outlier(s) to a trimmed partial sum of the sample. They are designed to enhance the robustness of outlier detection in samples with exponential or Pareto tails. These statistics are invariant to scale parameters and offer improved overall resistance to masking and swamping by recalibrating the denominator to reduce the influence of the largest observations. In particular, the proposed tests are shown to substantially reduce the masking problem in inward sequential testing, thereby re-establishing the inward sequential testing method - formerly relegated since the introduction of outward testing - as a competitive alternative to outward testing, without requiring multiple testing correction. The analytical null distributions of the statistics are derived, and a comprehensive comparison of the test statistics is conducted through simulation, evaluating the performance of the proposed tests in both block and sequential settings, and contrasting their performance with classical statistics across various data scenarios. In five case studies - financial crashes, nuclear power generation accidents, stock market returns, epidemic fatalities, and city sizes - significant outliers are detected and related to the concept of 'Dragon King' events, defined as meaningful outliers that arise from a unique generating mechanism.