# Chuan-Fa Tang (Department of Mathematical Sciences at University of Texas at Dallas)

Date

Wednesday March 30, 202210:00 am - 11:00 am

Location

Online via Zoom## Statistics Seminar

**Wednesday, March 30th, 2022**

**Time:** 10:00 a.m. **Place:** Online via Zoom (contact Brian Ling for Zoom link)

**Speaker:** Chuan-Fa Tang (Department of Mathematical Sciences at University of Texas at Dallas)

**Title:** Taylor's law for semivariance and higher moments of heavy-tailed distributions

**Abstract:** 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 distribution 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.