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

Spencer Wheatley, Didier Sornette|arXiv (Cornell University)|Jan 1, 2015

Complex Systems and Time Series AnalysisEconomics, Econometrics and Finance37 references2 citations

TL;DR

This paper proposes a robust inward sequential testing approach for detecting multiple outliers in exponential and Pareto-tailed samples, demonstrating it is as powerful as outward testing but simpler and less error-prone. It identifies 'Dragon King' events—meaningful, unique outliers—in real-world data, showing that correct distributional specification is critical for reliable inference.

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.

Motivation & Objective

• To address the limitations of outward testing in multiple outlier detection by proposing a more robust and reliable inward sequential approach.
• To evaluate the performance of various test statistics in block and sequential testing under different outlier configurations, including dispersed and clustered outliers.
• To explore the sensitivity of outlier detection to correct specification of the underlying exponential or Pareto distribution.
• To identify and analyze 'Dragon King' events—outliers of unique, meaningful origin—in real-world datasets such as financial crashes and epidemic fatalities.
• To demonstrate that inward testing avoids the complex type I error control required by outward tests, improving practical usability.

Proposed method

• Adapts common test statistics through a robust modification to enhance performance in the presence of multiple outliers.
• Employs inward sequential testing, where the most extreme observations are tested in ascending order of extremity, rather than starting from the most extreme.
• Compares block and sequential testing frameworks using a range of test statistics to evaluate power and error rates.
• Applies a density mixture model to detect clustered outliers, treating them as a distinct class from dispersed outliers.
• Uses Extreme Value Theory to justify the use of exponential and Pareto distributions for modeling tail behavior in general samples.
• Validates the method on five real-world case studies to identify Dragon King events through statistical significance and distributional fit.

Experimental results

Research questions

• RQ1 How does inward sequential testing compare in power and error rate to outward testing for detecting multiple outliers in exponential and Pareto samples?
• RQ2 What is the impact of incorrect distributional assumptions (exponential or Pareto) on outlier detection accuracy and inference reliability?
• RQ3 Can a robust modification of standard test statistics improve the performance of inward sequential testing?
• RQ4 How can clustered outliers—often overlooked—be effectively detected using statistical models?
• RQ5 Do significant outliers identified in real-world datasets (e.g., financial crashes, epidemic fatalities) qualify as 'Dragon King' events of unique origin?

Key findings

• The robust inward sequential testing approach achieves power comparable to outward testing while avoiding the need for complex type I error control.
• Inward testing is less prone to error and more practical for real-world applications due to its simplicity and stability.
• Outlier detection performance is highly sensitive to correct specification of the main distribution; misspecification leads to substantial inference errors.
• The density mixture model effectively detects clustered outliers, which are often missed by classical slippage models.
• Five real-world case studies—financial crashes, nuclear accidents, stock returns, epidemic fatalities, and city size distributions—revealed significant outliers consistent with the Dragon King concept.
• The detected outliers in these case studies were not random extremes but had unique, identifiable causes, supporting the Dragon King hypothesis of meaningful, high-impact events.