Statistics and Its Interface

Volume 11 (2018)

Number 4

Jackknife empirical likelihood for the skewness and kurtosis

Pages: 709 – 719



Yichuan Zhao (Georgia State University. Atlanta, Ga., U.S.A.)

Anna Moss (Georgia State University. Atlanta, Ga., U.S.A.)

Hanfang Yang (Renmin University of China, Beijing, China)

Yan Zhang (LexisNexis Risk Solutions, U.S.A.)


Coefficients of skewness and kurtosis provide convenient measures for describing the shape of a distribution based on a sample of independent observations. In this paper, we propose jackknife empirical likelihood (JEL) confidence intervals for the skewness and kurtosis coefficients, proving that the limiting distribution of the JEL ratio is a standard chi-squared distribution, and conduct an extensive simulation study comparing JEL with bootstrap methods. Compared with bootstrap methods, the JEL-based confidence intervals perform well in simulations with data from normal, $t$, $\gamma$, log-normal, and uniform distributions. We also illustrate the application of our proposed JEL methods using data from the behavioral risk factor surveillance system, an annual US health survey.


empirical likelihood, jackknife pseudo-values, kurtosis and skewness, nonparametric confidence intervals, U-statistics

Yichuan Zhao is supported by National Science Foundation(NSF) grant DMS-1613176, and by National Security Agency (NSA) grant H98230-12-1-0209. Hanfang Yang is supported by the National Natural Science Foundation of China, grant no. 11501567.

Received 6 February 2017

Published 19 September 2018