Applying the shadow distribution approach developed by the authors for violent conflicts , we provide rough estimates for quantities not immediately observable in the data.
Epidemics and pandemics are extremely heavy-tailed, with a potential existential risk for humanity. This property should override conclusions derived from local epidemiological models in what relates to tail events.
Empirical distributions have their in-sample maxima as natural censoring. We look at the "hidden tail", that is, the part of the distribution in excess of the maximum for a sample size of n. Using extreme value theory, we examine the properties of the hidden tail and calculate its moments of order p.
The method is useful in showing how large a bias one can expect, for a given n, between the visible in-sample mean and the true statistical mean (or higher moments), which is considerable for α close to 1.
Among other properties, we note that the "hidden" moment of order 0, that is, the exceedance probabil- ity for power law distributions, follows an exponential distribution and has for expectation 1/n regardless of the parametrization of the scale and tail index.
The book 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=∞, 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.