*“Vilfredo Pareto noticed that 80% of the land in Italy belonged to 20% of the population, and vice-versa, thus both giving birth to the power law class of distributions and the popular saying 80/20.”*

**Y**esterday, in “one of those” coincidences, I voluntarily dropped Nassim Taleb’s *The Bed of Procrustes* in a suburban café as my latest contribution to the book-crossing (or bXing!) concept and spotted a newly arXived paper by Taleb and Douadi. Paper which full title is *“On the Biases and Variability in the Estimation of Concentration Using Bracketed Quantile Contributions”* and which central idea is that estimating

(where q_{α} is the α-level quantile of X) by the ratio

can be strongly biased. And that the fatter the tail (i.e. the lower the power β for a power law tail), the worse the bias. This is definitely correct, if not entirely surprising given that the estimating ratio involves a ratio of estimators, plus an estimator of q_{α}. And that both numerator and denominator have finite variances when the power β is less than 2. The paper contains a simulation experiment easily reproduced by the following R code

#biased estimator of kappa(.01)
alpha=.01 #tail
omalpha=1-alpha
T=10^4 #simulations
n=10^3 #sample size
beta=1.1 #Pareto parameter
moobeta=-1/beta
kap=rep(0,T)
for (t in 1:T){
sampl=runif(n)^moobeta
quanta=quantile(sampl,omalpha)
kap[t]=sum(sampl[sampl>quanta])/sum(sampl)
}

**W**hat is somewhat surprising though is that the paper deems it necessary to run T=10¹² simulations to assess the bias when this bias is already visible in the first digit of κ_{α}. Given that the simulation experiment goes as high as n=10⁸, this means the authors simulated 10²⁰ Pareto variables to exhibit a bias a few thousand replicas could have produced. Checking the numerators and denominators in the above collection of ratios also shows that they may take unbelievably large values.)

*“…some theories are built based on claims of such `increase’ in inequality, as in Piketti (2014), without taking into account the true nature of κ, and promulgating theories about the `variation’ of inequality without reference to the stochasticity of the* *estimation—and the lack of consistency of **κ* across time and sub-units.”

**T**he more relevant questions about this issue of estimating κ_{α} are, in my opinion, (a) why this quantity is of enough practical importance to consider its estimation and to seek estimators that would remain robust as the power β varies arbitrarily close to 1; (b) in which sense there is anything more to the phenomenon than the difficulty in estimating β itself; and (c) what is the efficient asymptotic variance for estimating κ_{α} (since there is no particular reason to only consider the most natural estimator). Despite the above quote, that the paper constitutes a major refutation of Piketty’s *Capital in the Twenty-First Century* is rather unlikely!

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