Effect of kurtosis on efficiency of some multivariate medians

Abstract

Up to now, various multivariate medians have been proposed. To support their applications, we study the effect of kurtosis on efficiency of some well-known multivariate medians. Results are established for the coordinatewise median, the spatial median, the Oja median, and their modified versions. Such results provide a basis for choosing among the multivariate medians in practical analyses. The effect of dimension on efficiency of those multivariate medians is also studied. It is found that the result of Brown [(1983), ‘Statistical Uses of the Spatial Median’, Journal of the Royal Statistical Society, Series B, 45, 25–30] for the spatial median and spherically symmetric normal distributions can be extended to some other medians and some non-normal distributions but does not hold in general.

Keywords:

• multivariate kurtosis
• multivariate median
• multivariate sign test
• asymptotic relative efficiency
• elliptically symmetric distribution

AMS 2000 Subject Classification:

• Primary: 62G05
• Secondary: 62H05

Acknowledgements

Many thanks are due to two anonymous referees, an Associate Editor and the editor for their insightful comments and valuable suggestions, which led to considerable improvements of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.