Trimmed and winsorized standard deviations based on a scaled deviation

Abstract

Trimmed (and winsorized) standard deviations based on a scaled deviation are introduced and studied. The influence functions and limiting distributions are obtained. The performance of the estimators with respect to high breakdown scale estimators is evaluated and compared. Unlike other high breakdown estimators which perform poorly for light-tailed distribution and when points near the centre are contaminated, the resulting trimmed (and winsorized) standard deviations are much more efficient than their predecessors at light-tailed distributions by suitably choosing the cutting parameter and highly efficient for heavy-tailed and skewed distributions. At the same time, they share the best breakdown point robustness of the sample median absolute deviation for any common trimming thresholds. Compared with their predecessors, they can achieve the best efficiency when the contaminating points are presented from areas around the centre. Indeed, the scaled-deviation-trimmed (winsorized) standard deviations behave very well overall and, consequently, represent very favourable alternatives to existing scale estimators.

Keywords:

• trimmed standard deviation
• winsorized standard deviation
• influence function
• breakdown point
• robustness
• efficiency
• asymptotic representation
• limiting distribution

Acknowledgements

I thank the Editor and two anonymous referees for their comments and suggestions, especially for bringing the paper by Lopuhaä 3 to our attention. The research is partially supported by NSF grants DMS-0234078 and DMS-0501174.