As one of the most fundamental problems in statistics, robust location estimation has many prominent solutions, such as the symmetric trimmed mean, symmetric Winsorized mean, Hodges–Lehmann estimator, Huber M-estimator, and median of means. Recent studies suggest that their biases concerning the mean can be quite different in asymmetric distributions, but the underlying mechanisms largely remain unclear. This study exploited a semiparametric method to classify distributions by the asymptotic orderliness of location estimates with varying breakdown points, showing their interrelations and connections to parametric distributions. Further deductions explain why the Winsorized mean typically has smaller biases compared to the trimmed mean; two sequences of semiparametric robust mean estimators emerge. Building on the $\gamma$-$U$-orderliness, the superiority of the median Hodges–Lehmann mean is discussed.

These papers are prepared for PNAS. The related codes and drafts were shared and have been publically posted on my Github one year ago. I am introducing this work in YouTube and Quora, if you are interested, please visit: https://www.youtube.com/@Iobiomathematics or https://www.quora.com/profile/Tuobang-Li-1/answers or https://www.researchgate.net/profile/Tuobang-Li-2. For more information, please visit https//github.com/tubanlee. Also, feel free to share it or contact tl@biomathematics.org, for more materials available by request.