Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 980153, 20 pages
Research Article

Characterizing Tukey β„Ž and β„Ž β„Ž -Distributions through 𝐿 -Moments and the 𝐿 -Correlation

Section on Statistics and Measurement, Department EPSE, Southern Illinois University Carbondale, P.O. Box 4618, 222-J Wham Building, Carbondale, IL 62901-4618, USA

Received 10 October 2011; Accepted 31 October 2011

Academic Editors: M. Cho and K. Karamanos

Copyright © 2012 Todd C. Headrick and Mohan D. Pant. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper introduces the Tukey family of symmetric β„Ž and asymmetric β„Ž β„Ž -distributions in the contexts of univariate 𝐿 -moments and the 𝐿 -correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of 𝐿 -skew, 𝐿 -kurtosis, and 𝐿 -correlations. The procedure can be applied in a variety of settings such as modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of 𝐿 -skew, 𝐿 -kurtosis, and 𝐿 -correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern.