Nonparametric Confidence Regions for L-Moments

Abstract

Methods for constructing joint confidence regions for L-skewness and L-kurtosis are compared by Monte Carlo simulation. Exact computations can be based on variance estimators given by Elamir and Seheult (2003, Journal of Statistical Planning and Inference) and by Wang and Hutson (2013, Journal of Applied Statistics). Confidence regions can also be constructed using the bootstrap; several variants are considered. The principal conclusions are that all methods perform poorly for heavy-tailed distributions , and that even for light-tailed distributions a sample size of 200 may be required in order to achieve good agreement between nominal and actual coverage probabilities. A bootstrap method based on estimation of the covariance matrix of the sample L-moment ratios is overall the best simple choice. Among the practical results is an L-moment ratio diagram on which confidence regions for sample L-moment statistics are plotted. This gives an immediate visual indication of whether different samples can be regarded as having been drawn from the same distribution, and of which distributions are appropriate for fitting to a given data sample.