Nassim Nicholas Taleb

Black Swans and the Domains of Statistics

(2007). Black Swans and the Domains of Statistics. The American Statistician: Vol. 61, No. 3, pp. 198-200.

A New Heuristic Measure of Fragility and Tail Risks: Application to Stress Testing

This paper presents a simple heuristic measure of tail risk, which is applied to individual bank stress tests and to public debt. Stress testing can be seen as a first order test of the level of potential negative outcomes in response to tail shocks. However, the results of stress testing can be misleading in the presence of model error and the uncertainty attending parameters and their estimation. The heuristic can be seen as a second order stress test to detect nonlinearities in the tails that can lead to fragility, i.e., provide additional information on the robustness of stress tests. It also shows how the measure can be used to assess the robustness of public debt forecasts, an important issue in many countries. The heuristic measure outlined here can be used in a variety of situations to ascertain an ordinal ranking of fragility to tail risks.

On the statistical properties and tail risk of violent conflicts

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

On the Super-Additivity and Estimation Biases of Quantile Contributions

Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to sample size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of the exponent. These estimators can vary over time and increase with the population size, as shown in this article, thus providing the illusion of structural changes in concentration. They are also inconsistent under aggregation and mixing distributions, as the weighted average of concentration measures for A and B will tend to be lower than that from A U B. In addition, it can be shown that under such fat tails, increases in the total sum need to be accompanied by increased sample size of the concentration measurement. We examine the estimation superadditivity and bias under homogeneous and mixed distributions.

Expected Shortfall Estimation for Apparently Infinite-Mean Models of Operational Risk

Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment. Introducing what we call the dual distribution, we show how to deal with heavy-tailed phenomena with a remote yet finite upper bound. We provide methods to compute relevant tail quantities such as the Expected Shortfall, which is not available under infinite-mean models, allowing adequate provisioning and capital allocation. This also permits a measurement of fragility. The main difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution. Our methodology is useful with apparently infinite-mean phenomena, as in the case of operational risk, but it can be applied in all those situations involving extreme fat tails and bounded support.

Gini estimation under infinite variance

We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index

On the Shadow Moments of Apparently Infinite-Mean Phenomena

\alpha\in(1,2)

Election predictions as martingales: an arbitrage approach

). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods, because of a downward bias that emerges under fat tails. This has important implications for the ongoing discussion about economic inequality. We start by discussing how the nonparametric estimator of the Gini index undergoes a phase transition in the symmetry structure of its asymptotic distribution, as the data distribution shifts from the domain of attraction of a light-tailed distribution to that of a fat-tailed one, especially in the case of infinite variance. We also show how the nonparametric Gini bias increases with lower values of

Gini Estimation Under Infinite Variance

\alpha

The Decline of Violent Conflicts: What Do the Data Really Say?

. We then prove that maximum likelihood estimation outperforms nonparametric methods, requiring a much smaller sample size to reach efficiency. Finally, for fat-tailed data, we provide a simple correction mechanism to the small sample bias of the nonparametric estimator based on the distance between the mode and the mean of its asymptotic distribution.