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Nonparametric confidence regions for $$L$$-moments. (English) Zbl 1337.62098
Choudhary, Pankaj K. (ed.) et al., Ordered data analysis, modeling and health research methods. In honor of H. N. Nagaraja’s 60th birthday. Selected papers based on the presentations at the international conference, Austin, TX, USA, March 7–9, 2014. Cham: Springer (ISBN 978-3-319-25431-9/hbk; 978-3-319-25433-3/ebook). Springer Proceedings in Mathematics & Statistics 149, 39-53 (2015).
Summary: 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 E. A. H. Elamir and A. H. Seheult [J. Stat. Plann. Inference 124, No. 2, 337–359 (2004; Zbl 1074.62024)] and by D. Wang and A. D. Hutson [“Joint confidence region estimation of L-moment ratios with an extension to right censored data”, J. Appl. Stat. 40, No. 2, 368–379 (2013; doi:10.1080/02664763.2012.744386)]. 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.
For the entire collection see [Zbl 1337.92005].
##### MSC:
 62G15 Nonparametric tolerance and confidence regions 62G09 Nonparametric statistical resampling methods 62G32 Statistics of extreme values; tail inference
##### Keywords:
bootstrap; kurtosis; maximum entropy; order statistics; skewness
LMOMENTS; lmomco
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