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November, 1976 Descriptive Statistics for Nonparametric Models. III. Dispersion
P. J. Bickel, E. L. Lehmann
Ann. Statist. 4(6): 1139-1158 (November, 1976). DOI: 10.1214/aos/1176343648


Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive lower bounds for the efficiency (over the family of all symmetric distributions for which the measures are defined) exist if the trimmed standard deviations are replaced by $p$th power deviations. However, these latter measures are no longer robust, although for $p < 2$ they are more robust than the standard deviation. The results of the paper suggest that a positive bound to the efficiency may be incompatible with robustness but that trimmed standard deviations and $p$th power deviations for $p = 1$ or 1.5 are quite satisfactory in practice.


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P. J. Bickel. E. L. Lehmann. "Descriptive Statistics for Nonparametric Models. III. Dispersion." Ann. Statist. 4 (6) 1139 - 1158, November, 1976.


Published: November, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0351.62031
MathSciNet: MR474620
Digital Object Identifier: 10.1214/aos/1176343648

Primary: 62G05
Secondary: 62G20 , 62G30 , 62G35

Keywords: $p$th power deviation , Asymptotic efficiency , dispersion , estimation , scaling a positive distribution , standard deviation , standardized asymptotic variance , substitution of an estimate of location , Tukey model

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 6 • November, 1976
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