Estimating Power Law Exponents For Extreme Capital Losses Without Probability Scoring
Power law tail exponents measure drawdown severity scaling directly from ordered loss ratios, bypassing subjective probability scoring in fat-tailed domains.
sample variance degeneration
Meaning
Statistical instabilities where the measured variance of a dataset fails to converge or increases as more data points are added indicate the presence of infinite variance. When sample variance degeneration occurs, the standard tools of statistics become misleading. This situation typically arises in systems governed by fat tails or Cauchy distributions.
The more observations one collects, the more likely one is to encounter an outlier that shifts the variance to a higher level.
Convergence Fault
Stability of the average is also threatened when the variance does not settle. In a standard system, the law of large numbers ensures that the sample mean eventually matches the population mean. Under the conditions of sample variance degeneration, the mean can jump wildly even after thousands of observations.
This makes it impossible to find a reliable baseline for performance or risk.
Data Instability
Reliance on historical averages leads to a false sense of precision. Because sample variance degeneration prevents the variance from reaching a fixed number, the error bars around a forecast actually grow as more data is included. This contradicts the common belief that more information always leads to better decisions.
Practitioners must use alternative measures, such as the mean absolute deviation, which may be more stable.
Estimation Error
Traditional models often ignore this issue by assuming that all data comes from a normal distribution. If the underlying process is prone to sample variance degeneration, the resulting risk estimates will be off by several orders of magnitude. The fault lies in the assumption that the system has a finite and measurable variance.