Multiple Outlier Detection in Samples with Exponential & Pareto Tails: Redeeming the Inward Approach & Detecting Dragon Kings

We introduce two ratio-based robust test statistics, max-robust-sum (MRS) and sum-robust-sum (SRS), designed to enhance the robustness of outlier detection in samples with exponential or Pareto tails. We also reintroduce the inward sequential testing method -- formerly relegated since the introduction of outward testing -- and show that MRS and SRS tests reduce susceptibility of the inward approach to masking, making the inward test as powerful as, and potentially less error-prone than, outward tests. Moreover, inward testing does not require the complicated type I error control of outward tests. A comprehensive comparison of the test statistics is done, considering performance of the proposed tests in both block and sequential tests, and contrasting their performance with classical test 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.