Robust measures of circular dispersion

G.C. Porzio

{}^{1}

H. Demni

{}^{1}

M. Hubert

{}^{2}

and P. Rousseeuw

{}^{2}

${}^{1}$

University of Cassino and Southern Lazio, Cassino, Italy [porzio, houyem.demni@unicas.it]
${}^{2}$

KU Leuven, Leuven, Belgium [mia.hubert@kuleuven.be, peter@rousseeuw.net]

Abstract

Circular variables indicate periodic observations or directions. They are of interest within the fields of biological and environmental sciences, Earth sciences, and astronomy, to cite a few.

An important issue to consider when dealing with circular data is how to measure their dispersion. This work addresses such an issue from a robust statistical perspective.

Particularly, it considers three well-worn robust measures of dispersion for the analysis of linear data (the Median Absolute Deviation, the Least Median Spread, and the Least Trimmed Spread), and it discusses their extension to the case of circular variables. Their properties are studied, including their robustness in terms of relative bias.

Finally, exploiting their relationship with the concentration parameter $\kappa$ of the von Mises distribution, they are utilized to define three new robust estimators of such a $\kappa$ . The estimator’s breakdown point was then investigated.

Because of the peculiarity of circular data, unlike the Median Absolute Deviation when used to estimate the variance of a distribution, we found that the breakdown value of the estimator based on the circular Median Absolute Deviation does not achieve the maximum attainable value of 0.5. On the other hand, the breakdown values of these estimators are generally satisfactory, and the maximum attainable value is proved to be achieved by the circular Least Median and Least Trimmed Spread based estimators.