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This book focuses on general frameworks for modeling heavy-tailed distributions in economics, finance, econometrics, statistics, risk management and insurance. A central theme is that of (non-)robustness, i.e., the fact that the presence of heavy tails can either reinforce or reverse the implications of a number of models in these fields, depending on the degree of heavy-tailedness. These results motivate the development and applications of robust inference approaches under heavy tails, heterogeneity and dependence in observations. Several recently developed robust inference approaches are discussed and illustrated, together with applications.

Statistics. --- Statistics for Business/Economics/Mathematical Finance/Insurance. --- Statistical Theory and Methods. --- Econometrics. --- Mathematical statistics. --- Economics --- Statistique --- Statistique mathématique --- Econométrie --- Distribution (Probability theory). --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Distribution (Probability theory) --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistics for Business, Management, Economics, Finance, Insurance. --- Economics, Mathematical --- Statistics --- Statistical inference --- Statistics, Mathematical --- Sampling (Statistics) --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Statistics .

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Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank)=a-b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank-1/2, and run log(Rank-1/2)=a-b log(Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent zeta is not the OLS standard error, but is asymptotically (2/n)^(1/2) zeta. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the U.S. city size distribution.