Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

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 (2−α)/(1−α)>2. 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 (2−α2)/(1−α). 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.

Heavy-tailed nonnegative random variables with infinite moments, such as nonnegative stable laws with index α in (0,1), have theoretical and practical importance e.g., Carmona (1), Feller (2), Resnick (3), and Samorodnitsky and Taqqu (4)]. Heavy-tailed nonnegative random variables with some or all infinite moments have been claimed to arise empirically in finance operational risks in Nešlehová et al. (5)], economics income distributions in Campolieti (6) and Schluter (7); returns to technological innovations in Scherer et al. (8) and Silverberg and Verspagen (9)], demography city sizes in Cen (10)], linguistics word frequencies in Bérubé et al. (11)], and insurance economic losses from earthquakes in Embrechts et al. (12) and Ibragimov et al. (13)]. Partial reviews are in Carmona (1) and Ibragimov (14).Brown et al. (15) (hereafter BCD) showed that when a random sample is drawn from a nonnegative stable law with index α∈(0,1), the sample variance is asymptotically (as the sample size n goes to ∞) proportional to the sample mean raised to a power that is an explicit function of α (Eqs. 11 and 13). This relationship generalizes to stable laws with infinite moments a widely observed power-law relationship between the variance and the mean in families of distributions with finite population mean and finite population variance. This power-law relationship is commonly known as Taylor’s law in ecology Taylor (16, 17)] and as fluctuation scaling in physics Eisler et al. (18)].To the two ingredients combined by BCD (nonnegative stable laws with infinite moments and Taylor’s law), this paper adds two more ingredients. We establish scaling relationships that generalize the usual Taylor’s law, for light-tailed distributions, to many functions of the sample in addition to the variance, including all positive absolute and central moments, upper and lower semivariances, and several measures of risk-adjusted investment performance such as the Sortino, Sharpe, and Farinelli–Tibiletti ratios. In addition, based on these scaling relationships, we propose several estimators of the index α of a nonnegative stable law with infinite first moment.Section 1 defines most of the sample functions studied here. Section 2 gives background on Taylor’s law, semivariances, and nonnegative stable laws, including key prior results from BCD. Section 3 establishes that the lower sample semivariance, the upper sample semivariance, the local lower sample semivariance, and the local upper sample semivariance are asymptotically each a power of the sample mean with explicitly given exponents. These results are the core of the paper. When investment returns obey a nonnegative heavy-tailed law with index α∈(0,1), these results reveal the asymptotic behavior of the Sharpe ratio, the Sortino ratio, and the Farinelli–Tibiletti ratio. Section 4 extends these results to higher central and noncentral moments and various indices of volatility. Section 5 analyzes the number of observations from a stable law or an approximately stable (i.e., regularly varying) law that exceed the sample mean. Section 6 proposes and compares estimators of α by simulation. SI Appendix gives all proofs of results stated in the text and additional numerical simulations.