The BAB algorithm for computing the total least trimmed squares estimator

Abstract

Robust estimation in the errors-in-variables (EIV) model remains a difficult problem because of the leverage point and the masking effect and swamping effect. In this contribution, a new robust estimator is introduced for the EIV model. This method is a follow-up to least trimmed squares, which is applied to the Gauss–Markov model when only the observation vector contains outliers. We call this estimator the total least trimmed squares (TLTS) estimator because its criterion function consists of squared orthogonal residuals. The TLTS estimator excludes some large squared orthogonal residuals from the criterion function, thereby allowing the fit to ignore outliers. The TLTS estimator inherits appropriate equivariance properties, namely regression equivariance, scale equivariance and affine equivariance, and the maximal 50% asymptotic breakdown point in terms of observations $$ \varvec{y} $$ within the special cofactor matrix structure. The TLTS estimate can directly be obtained by the exhaustive evaluation method. We further develop another algorithm for the TLTS estimator based on the branch-and-bound method without exhaustive evaluation, but the cofactor matrix of the independent variables needs to have a certain block structure. Finally, two simulation studies provide insights into the robustness and efficiency of the proposed algorithms.