Robustness of Msplit(q) estimation: A theoretical approach

M-split(q) estimation is a development of M-estimation which is based on the assumption that a functional model of observations can be split into q competitive ones. The main idea behind such an assumption is that the observation set might be a mixture of realizations of different random variables which differ from each other in location parameters that are estimated. The paper is focused on the robustness of M-split(q) estimates against outlying observations. The paper presents derivatives of the general expressions of the respective influence functions and weight functions which are the main basis for theoretical analysis. To recognize the properties of M-split(q) estimates in a better way, we propose considering robustness from two points of view, namely local and global ones. Such an approach is a new one, but it reflects the nature of the estimation method in question very well. Thus, we consider the local breakdown point (LBdP) and the global one (GBdP) that are both based on the maximum sensitivities of the estimates. LBdP describes the mutual relationship between the "neighboring" M-split(q) estimates, whereas GBdP concerns the whole set of the estimates and describes the robustness of the method itself (in more traditional sense). The paper also presents GBdP with an extension, which shows how an outlier might influence M-split(q) estimates. The general theory proposed in the paper is applied to investigate the squared M-split(q) estimation, the variant which is used in some practical problems in geodesy, surveying, remote sensing or geostatistics, and which can also be applied in other geosciences.