Least sum of squares of trimmed residuals regression

In the famous least sum of trimmed squares (LTS) of residuals estimator (Rousseeuw (1984)), residuals are first squared and then trimmed. In this article, we first trim residuals - using a depth trimming scheme - and then square the rest of residuals. The estimator that can minimize the sum of squares of the trimmed residuals, is called an LST estimator. It turns out that LST is also a robust alternative to the classic least sum of squares (LS) of residuals estimator. Indeed, it has a very 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 LST is Fisher consistent, and the sample version is strong and root-n consistent under some conditions. Three approximate algorithms for computing LST are proposed and tested in synthetic and real data examples. These experiments indicate that two of the algorithms can compute the LST estimator very fast and with relatively smaller variances, compared with that of the famous LTS estimator. All the evidence suggests that LST deserves to be a robust alternative to the LS estimator and is feasible in practice for large data sets (with possible contamination and outliers) in high dimensions.