Sign-Perturbed Sums (SPS) with asymmetric noise: Robustness analysis and robustification techniques

Sign-Perturbed Sums (SPS) is a recently developed finite sample system identification method that can build exact confidence regions for linear regression problems under mild statistical assumptions. The regions are well-shaped, e.g., they are centred around the least-squares (LS) estimate, star-convex and strongly consistent. One of the main assumptions of SPS is that the distribution of the noise terms are symmetric about zero. This paper analyses how robust SPS is with respect to the violation of this assumption and how it could be robustified with respect to non-symmetric noises. First, some alternative solutions are overviewed, then a robustness analysis is performed resulting in a robustified version of SPS. We also suggest a modification of SPS, called LAD-SPS, which builds exact confidence regions around the least-absolute deviation (LAD) estimate instead of the LS estimate. LAD-SPS requires less assumptions as the noise needs only to have a conditionally zero median (w.r.t. the past). Furthermore, that approach can also be robustified using similar ideas as in the LS-SPS case. Finally, some numerical experiments are presented.