November 12, 2021
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Taylor's law of fluctuation scaling for semivariances and higher moments of heavy-tailed data - pnas.org

Significance

Many quantities are extremely large extremely rarely. Examples include income, wealth, financial returns, insurance losses, firm size, and city population size; earthquake magnitude, hurricane energy, tornado outbreaks, precipitation, and flooding; and pest outbreaks, infectious epidemics, and forest fires. When such a quantity is modeled as a nonnegative random variable with a heavy upper tail, the probability of an observation larger than some threshold falls as a small power (the “tail index”) of the threshold. When the tail index is small enough, the mean and all higher moments of the random quantity are infinite. Surprisingly, the sample mean and the sample higher moments obey orderly scaling laws, which we prove and apply to estimating the tail index.

Abstract

We generalize Taylor’s law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power . The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power . These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli–Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor’s law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Footnotes

  • Author contributions: M.B., J.E.C., C.-F.T., and S.C.P.Y. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

  • Reviewers: S.R., Texas Tech University; and J.R., London School of Economics.

Computer code has been deposited in GitHub (https://github.com/cftang9/TLHM). Readers can generate the tables and figures using the R code there.



source: https://www.pnas.org/content/118/46/e2108031118

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