Nassim Taleb – Simplest Forms
One-Tailed
- 1-2 Parameter: Minimum value, tail (Pareto I)
- 3 Parameters: location m, scale s, tail Alpha (Pareto 2-Lomax)
- 4 Parameters: Generalized Beta 2nd kind (includes many known distributions such as Singh-Maddala, etc.)
Two-Tailed
- Student T distribution (finance papers galore)
- Levy-Stable (to which all those with Alpha<2 converge)
- Other (double Pareto, etc.)
Sample Equivalence
Dispersion of outcomes: for exponent <2, we cannot use standard deviation and other tools since no second moment. Only MAD , mean absolute deviation of the mean from “true” mean (or 0).
Indexing by p for powerlaw and g Gaussian:
Back door working
- Sample mean (“realized a average” in language of finance) is never Gaussian when Alpha<2 (even when >2, another story on CLT (following chapter in Silent Risk))
- Tail Alpha can be estimated with MLE
- Tail Alpha from MLE is asymptotically Gaussian (preasymptotically Inverse Gamma with low variance reaches v. quickly)
- We can fit GPD or EVD with same tail exponent Alpha, further reducing variance.
A few points
When 1<Alpha<2 we can safely say that the sample mean is insignificant and underestimates the mean for one-tailed distributions (reason: infinite skewness). For all sample size.
In some cases, with Alpha<1, we can extract the “true” mean, albeit stochastic. Case study on violence (Cirillo and Taleb, 2015).
Application: the Pikery craze.
Centile contribution: is Pareto 80/20 true?
80/20 by recursing -> Top 1% has 53%
Huuuuuuuuge bias in mean measurement as bracketed -> y-o-y changes suspicious
Conclusion
- Much of finance, social science, relies on bogus estimators. For instance Pinker’s problem is quite insidious with mechanistic users of statistics.
- We “recalibrate” models because they are not estimators, chasing past fitness. – As the late Benoit Mandelbrot said: when a lightning hits we do not change the laws of nature.
- Excellent news: rigorous methods, including using extreme value theory and development of new estimators clears up a lot of problems
See full slides below.
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