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

Abstract

We consider the detection of multiple outliers in Exponential and Pareto samples -- as well as general samples that have approximately Exponential or Pareto tails, thanks to Extreme Value Theory. It is shown that a simple robust'' modification of common test statistics makes inward sequential testing -- formerly relegated within the literature since the introduction of outward testing -- as powerful as, and potentially less error prone than, outward tests. Moreover, inward testing does not require the complicated type 1 error control of outward tests. A variety of test statistics, employed in both block and sequential tests, are compared for their power and errors, in cases including no outliers, dispersed outliers (the classical slippage alternative), and clustered outliers (a case seldom considered). We advocate a density mixture approach for detecting clustered outliers. Tests are found to be highly sensitive to the correct specification of the main distribution (Exponential/Pareto), exposing high potential for errors in inference. Further, in five case studies -- financial crashes, nuclear power generation accidents, stock market returns, epidemic fatalities, and cities within countries -- significant outliers are detected and related to the concept of ‘Dragon King’ events, defined as meaningful outliers of unique origin.