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On the Super-Additivity and Estimation Biases of Quantile Contributions

Physica A: Statistical Mechanics and Applications, Forthcoming

NYU Tandon Research Paper No. 2434363

6 Pages Posted: 9 May 2014 Last revised: 27 Jun 2017

Nassim Nicholas Taleb

NYU-Tandon School of Engineering

Raphael Douady

Riskdata; Stony Brook university ; CES Univ. Paris 1

Date Written: November 11, 2014


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.

Keywords: Risk, Inequality, Statistics

Suggested Citation

Taleb, Nassim Nicholas and Douady, Raphael, On the Super-Additivity and Estimation Biases of Quantile Contributions (November 11, 2014). Physica A: Statistical Mechanics and Applications, Forthcoming; NYU Tandon Research Paper No. 2434363. Available at SSRN: or

Nassim Nicholas Taleb (Contact Author)

NYU-Tandon School of Engineering ( email )

Bobst Library, E-resource Acquisitions
20 Cooper Square 3rd Floor
New York, NY 10003-711
United States

Raphael Douady

Riskdata ( email )

6, rue de l'Amiral Coligny
Paris, 75001


Stony Brook university ( email )

Stony Brook, NY 11794
United States

CES Univ. Paris 1 ( email )

106 bv de l'Hôpital
Paris, 75013

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