Tolerance for Growing Errors of Observations as a Measure Describing Global Robustness of $M_{\mathrm{split}}$ Estimation and Providing New Information on Other Methods

$M_{\mathrm{split}}$ estimation is a modern estimation method that has found various applications in processing geodetic data. Its basic variants were not meant to be robust against outliers; however, the practical applications showed that the method could be used in such a context. Therefore, there is a need to describe the robustness of different $M_{\mathrm{split}}$ estimation variants. The paper uses the global breakdown point in an extended interval (GBdP-$e$) but also introduces the tolerance for growing errors of observations (TGE) to perform such an examination. It presents such measures obtained for the absolute $M_{\mathrm{split}}$ estimation and robust $M_{\mathrm{split}}$ estimation variants, which have not been shown before. The results prove that the absolute $M_{\mathrm{split}}$ estimation predominates the squared $M_{\mathrm{split}}$ estimation in such a context. Furthermore, the robust variants are much less sensitive to outliers than both basic variants mentioned. TGE not only describes how the method tolerates outliers but could also be applied to assume the most appropriate values of the steering parameters, which seems essential. The paper shows the theoretical relationship between basic $M_{\mathrm{split}}$ estimation variants and respective $M$-estimation methods. It is a basis for introducing and deriving GBdP-$e$ and also TGE for $M$-estimation. The paper shows that both measures are equivalent in the case of $M$-estimation. TGE could provide information about that estimation type’s sensitivity to growing errors of observations (also robustness to outliers) that is unavailable by applying other measures, including classical breakdown points, influence functions, rejection points, or mean success rate. TGE presents the robustness potential of the $M$-estimation variants in a rather vivid and straightforward way, even for methods not classified as robust against outliers, e.g., the least-squares estimation.