Open Access
2023 Least sum of squares of trimmed residuals regression
Yijun Zuo, Hanwen Zuo
Author Affiliations +
Electron. J. Statist. 17(2): 2416-2446 (2023). DOI: 10.1214/23-EJS2164


In the famous least sum of trimmed squares (LTS) estimator [21], residuals are first squared and then trimmed. In this article, we first trim residuals – using a depth trimming scheme – and then square the remaining of residuals. The estimator that minimizes the sum of trimmed and squared residuals, is called an LST estimator.

Not only is the LST a robust alternative to the classic least sum of squares (LS) estimator. It also has a high finite sample breakdown point-and can resist, asymptotically, up to 50% contamination without breakdown – in sharp contrast to the 0% of the LS estimator.

The population version of the LST is Fisher consistent, and the sample version is strong, root-n consistent, and asymptotically normal. We propose approximate algorithms for computing the LST and test on synthetic and real data sets. Despite being approximate, one of the algorithms compute the LST estimator quickly with relatively small variances in contrast to the famous LTS estimator. Thus, evidence suggests the LST serves as a robust alternative to the LS estimator and is feasible even in high dimension data sets with contamination and outliers.

Funding Statement

Authors declare that there is no funding received for this study.


The authors thank Denis Selyuzhitsky, Nadav Langberg, and Professsors Wei Shao and Yimin Xiao for their stimulating discussions and the authors thank the Co-Editor-in-Chiefs, Professors Grace Y. Yi and Gang Li and the anonymous referees for their insightful and constructive comments. All of this feedback has significantly improved the manuscript.


Download Citation

Yijun Zuo. Hanwen Zuo. "Least sum of squares of trimmed residuals regression." Electron. J. Statist. 17 (2) 2416 - 2446, 2023.


Received: 1 November 2022; Published: 2023
First available in Project Euclid: 9 October 2023

arXiv: 2202.10329
MathSciNet: MR4651887
Digital Object Identifier: 10.1214/23-EJS2164

Primary: 62G36 , 62J05
Secondary: 62G99 , 62J99

Keywords: approximate computation algorithm , consistency , finite sample breakdown point , robust regression , Trimmed residuals

Vol.17 • No. 2 • 2023
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