Quantile absolute deviation

Abstract: The median absolute deviation (MAD) is a popular robust measure of statistical dispersion. However, when it is applied to non-parametric distributions (especially multimodal, discrete, or heavy-tailed), lots of statistical inference issues arise. Even when it is applied to distributions with slight deviations from normality and these issues are not actual, the Gaussian efficiency of the MAD is only 37% which is not always enough. In this paper, we introduce the quantile absolute deviation (QAD) as a generalization of the MAD. This measure of dispersion provides a flexible approach to analyzing properties of non-parametric distributions. It also allows controlling the trade-off between robustness and statistical efficiency. We use the trimmed Harrell-Davis median estimator based on the highest density interval of the given width as a complimentary median estimator that gives increased finite-sample Gaussian efficiency compared to the sample median and a breakdown point matched to the QAD. As a rule of thumb, we suggest using two new measures of dispersion called the standard QAD and the optimal QAD. They give 54% and 65% of Gaussian efficiency having breakdown points of 32% and 14% respectively.