### On The Statistical Properties And Tail Risk Of Violent Conflicts

Delft University of Technology; Delft University of Technology – Delft Institute of Applied Mathematics (DIAM)

Independent Researcher; NYU-Poly Institute; New England Complex Systems Institute

October 16, 2015

**Abstract: **

We examine statistical pictures of violent conflicts over the last 2000 years, finding techniques for dealing with incompleteness and unreliability of historical data.

We introduce a novel approach to apply extreme value theory to fat-tailed variables that have a remote, but nonetheless finite upper bound, by defining a corresponding unbounded dual distribution (given that potential war casualties are bounded by the world population).

We apply methods from extreme value theory on the dual distribution and derive its tail properties. The dual method allows us to calculate the real mean of war casualties, which proves to be considerably larger than the sample mean, meaning severe underestimation of the tail risks of conflicts from naive observation. We analyze the robustness of our results to errors in historical reports, taking into account the unreliability of accounts by historians and absence of critical data.

We study inter-arrival times between tail events and find that no particular trend can be asserted.

All the statistical pictures obtained are at variance with the prevailing claims about “long peace,” namely that violence has been declining over time.

### On The Statistical Properties And Tail Risk Of Violent Conflicts – Introduction

Since the middle of last century, there has been a multidisciplinary interest in wars and armed conflicts (quantified in terms of casualties). Studies have also covered the statistics of terrorism, for instance [8], [35], and the special issue of Risk Analysis on terrorism [33]. From a statistical point of view, recent contributions have attempted to show that the distribution of war casualties (or terrorist attacks’ victims) tends to have heavy tails, characterized by a power law decay [8] and [14]. Often, the analysis of armed conflicts falls within the broader study of violence [6], [30], with the aim to understand whether we as human are more or less violent and aggressive than in the past and what role institutions played in that respect. Accordingly, the public intellectual arena has witnessed active debates, such as the one between Steven Pinker on one side, and John Gray on the other concerning the hypothesis that the long peace was a statistically established phenomenon or a mere statistical sampling error that is characteristic of heavy-tailed processes, [16] and [27] –the latter of which is corroborated by this paper.

Using a more recent data set containing 565 armed conflicts with more than 3000 casualties over the period 1-2015 AD, we confirm that the distribution of war casualties exhibits a very heavy right-tail. The tail is so heavy that — at first glance — war casualties could represent an infinite-mean phenomenon, as defined by [26]. But should the distribution of war casualties have an infinite mean, the annihilation of the human species would be just a matter of time, and the sample properties we can compute from data have no meaning at all in terms of statistical inference. In reality, a simple argument allows us to rule out the infiniteness of the mean: no event or series of events can kill more than the whole world population. The support of the distribution of war casualties is necessarily bounded, and the true mean cannot be infinite.

By studying the tail properties of the dual distribution (the one with an infinite upper bound), using extreme value theory, we will be able to obtain, by reverting to the real distribution, what we call the shadow mean of war casualties. We will show that this mean is at least 1.5 times larger than the sample mean, but nevertheless finite.

We assume that many observations are missing from the dataset (from under-reported conflicts), and we base on analysis on the fact that war casualties are just imprecise estimates [37], on which historians often have disputes, without anyone’s ability to verify the assessments using period sources. For instance, an event that took place in the eighth century, the An Lushan rebellion, is thought to have killed 13 million people, but there no precise or reliable methodology to allow us to trust that number –which could be conceivably one order of magnitude smaller.2. Using resampling techniques, we show that our tail estimates are robust to changes in the quality and reliability of data. Our results and conclusions will replicate even we missed a third of the data. When focusing on the more reliable set covering the last 500 years of data, one cannot observe any specific trend in the number of conflicts, as large wars appear to follow a homogeneous Poisson process. This memorylessness in the data conflicts with the idea that war violence has declined over time, as proposed by [15] or [30].

The paper is organized as follows: Section I describes our data set and analyses the most significant problems with the quality of observations; Section III is a descriptive analysis of data; it shows some basic result, which are already sufficient to refute the thesis [15] that we are living in a more peaceful world on the basis of statistical observations; Section IV contains our investigations about the upper tail of the dual distribution of war casualties as well as discussions of tail risk; Section V deals with the estimation of the shadow mean; in Section VI we discuss the robustness of our results to imprecision and errors in the data; finally, Section VII looks at the number of conflicts over time, showing no visible trend in the last 500 years.

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