Least sum of squares of trimmed residuals regression

Abstract

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 $50\mathrm{\%}$ contamination without breakdown – in sharp contrast to the $0\mathrm{\%}$ 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.