Robust Statistical Tests of Dragon-Kings beyond Power Law Distributions

TL;DR

This paper introduces two new statistical tests, the U-test and DK-test, designed to detect anomalous 'Dragon-Kings' events in distributions, demonstrated on city and agglomeration data across several countries.

Contribution

The paper presents novel, distribution-independent tests for identifying single extreme events in small samples, advancing the detection of Dragon-Kings beyond power law models.

Findings

London, Moscow, St-Petersburg, and Paris show Dragon-Kings.

No Dragon-Kings found in German city data.

Tests effectively identify anomalous events in real-world distributions.

Abstract

We ask the question whether it is possible to diagnose the existence of "Dragon-Kings" (DK), namely anomalous observations compared to a power law background distribution of event sizes. We present two new statistical tests, the U-test and the DK-test, aimed at identifying the existence of even a single anomalous event in the tail of the distribution of just a few tens of observations. The DK-test in particular is derived such that the p-value of its statistic is independent of the exponent characterizing the null hypothesis. We demonstrate how to apply these two tests on the distributions of cities and of agglomerations in a number of countries. We find the following evidence for Dragon-Kings: London in the distribution of city sizes of Great Britain; Moscow and St-Petersburg in the distribution of city sizes in the Russian Federation; and Paris in the distribution of agglomeration sizes in France. True negatives are also reported, for instance the absence of Dragon-Kings in the distribution of cities in Germany.