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**Trimmed and Winsorized means based on a scaled deviation.**
*(English)*
Zbl 1149.62047

Summary: Trimmed (and Winsorized) means based on a scaled deviation are introduced and studied. The influence functions of the estimators are derived and their limiting distributions are established via asymptotic representations. As a main focus of the paper, the performance of the estimators with respect to various robustness and efficiency criteria is evaluated and compared with leading competitors including the ordinary Tukey trimmed (and Winsorized) means. Unlike the Tukey trimming which always trims a fixed fraction of sample points at each end of data, the trimming scheme here only trims points at one or both ends that have a scaled deviation beyond some threshold. The resulting trimmed (and Winsorized) means are much more robust than their predecessors.

Indeed they can share the best breakdown point robustness of the sample median for any common trimming thresholds. Furthermore, for appropriate trimming thresholds they are highly efficient at light-tailed symmetric models and more efficient than their predecessors at heavy-tailed or contaminated symmetric models. Detailed comparisons with leading competitors on various robustness and efficiency aspects reveal that the scaled deviation trimmed (Winsorized) means behave very well overall and consequently represent very favorable alternatives to the ordinary trimmed (Winsorized) means.

Indeed they can share the best breakdown point robustness of the sample median for any common trimming thresholds. Furthermore, for appropriate trimming thresholds they are highly efficient at light-tailed symmetric models and more efficient than their predecessors at heavy-tailed or contaminated symmetric models. Detailed comparisons with leading competitors on various robustness and efficiency aspects reveal that the scaled deviation trimmed (Winsorized) means behave very well overall and consequently represent very favorable alternatives to the ordinary trimmed (Winsorized) means.

### MSC:

62H12 | Estimation in multivariate analysis |

62G35 | Nonparametric robustness |

62E20 | Asymptotic distribution theory in statistics |

62G05 | Nonparametric estimation |

### Keywords:

trimmed mean; Winsorized mean; influence function; breakdown point; robustness; efficiency; asymptotic representation; limiting distribution
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\textit{M. Wu} and \textit{Y. Zuo}, J. Stat. Plann. Inference 139, No. 2, 350--365 (2009; Zbl 1149.62047)

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