Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications

TL;DR

This paper explores the statistical implications of fat-tailed distributions, highlighting how traditional methods often fail in real-world scenarios and proposing insights into more accurate analysis and modeling.

Contribution

It provides a comprehensive analysis of the misapplications of classical statistics to fat-tailed data and synthesizes recent research to improve understanding and methodology.

Findings

Sample mean often diverges from population mean in fat-tailed contexts

Empirical distributions are rarely truly empirical in practice

Standard statistical tools like principal components and inequality measures can be misleading

Abstract

(The third edition corrects minor typos and adds 3 chapters synthesized from published papers plus an appendix on maximum entropy distributions.) The monograph investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress". Traditional asymptotics deal mainly with either n=1 or $n=\infty$, and the real world is in between, under of the "laws of the medium numbers" --which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: + The sample mean is rarely in line with the population mean, with effect on "naive empiricism", but can be sometimes be estimated via parametric methods. + The "empirical distribution" is rarely empirical. + Parameter uncertainty has compounding effects on statistical metrics. + Dimension reduction (principal components) fails. + Inequality estimators (GINI or quantile contributions) are not additive and produce wrong results. + Many "biases" found in psychology become entirely rational under more sophisticated probability distributions + Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.