Taylor’s Law for Semivariance and Higher Moments of Heavy-Tailed Distributions

Presented by Chuan-Fa Tang, Assistant Professor, Department of Mathematical Sciences, University of Texas at Dallas

Wednesday, April 20, 2022
4:00 p.m. ET
Online

The power law relates the population mean and variance is known as Taylor’s law, proposed by Taylor in 1961. We generalize Taylor’s law from the light-tailed distributions to heavy-tailed distributions with infinite mean. Instead of population moments, we consider the power-law between the sample mean and many other sample statistics, such as the sample upper and lower semivariance, the skewness, the kurtosis, and higher moments of a random sample. We show that, as the sample size increases, the preceding sample statistics increase asymptotically in direct proportion to the power of the sample mean. These power 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 potential upside ratio, and the Farinelli-Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. In addition, we find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of tail-index based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators.

Speaker Bio

Dr. Chuan-Fa Tang is an Assistant Professor in the Department of Mathematical Sciences at the University of Texas at Dallas. His research interests include order-restricted inference, shaped-constrained inference, empirical processes, empirical likelihood, survival analysis, mathematical statistics, image processing, kernel smoothing, and model selection. Dr. Tang received his PhD in Statistics from the University of South Carolina in 2017.