Assessing Symmetry Using Quantiles and L-Moments

Abstract

Measures of distributional symmetry based on quantiles, L-moments, and trimmed L-moments are briefly reviewed, and (asymptotic) sampling properties of commonly used estimators considered. Standard errors are estimated using both analytical and computer-intensive methods. Simulation is used to assess results when sampling from some known distributions; bootstrapping is used on sample data to estimate standard errors, construct confidence intervals, and test a hypothesis of distributional symmetry. Symmetry measures based on 2- or 3-trimmed L-moments have some advantages over other measures in terms of their existence. Their estimators are generally well behaved, even in relatively small samples.

Keywords:

• Bootstrap estimation
• Bootstrap testing
• L-moments
• Quantiles
• Skewness
• Symmetry
• Trimmed L-moments

Mathematics Subject Classification:

• 62F
• 62F10
• 62F12
• 62F40

Acknowledgments

The author wishes to thank a referee and Dr. Mike Green, University of Dundee, for their valuable comments on earlier versions of this article.