Descriptive Statistics for Nonparametric Models. III. Dispersion

Abstract

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.