This paper presents a new method of estimating a shift between parameters of a functional model of observations. The method in question is based on the theory of Msplit estimation and assumes that design matrices of functional models may not have full column rank. Thus, this alternative method, which is called Shift-Msplit(+) estimation, can be applied to analyze deformations of free geodetic networks. The shift of the parameters can then be regarded as a shift between the coordinates of the network points at two different measurement epochs. Such an application is illustrated using a numerical example, and the results are compared with the results of R-estimation and with a conventional free adjustment of the least squares. This paper presents the theory of Shift-Msplit(+) estimators and ways to compute them. The procedures in question are based on the general principles of Msplit estimation and practical application of the Moore–Penrose inverses.

The paper concerns \(M\) -estimation with probabilistic models of geodetic observations that is called \(M_{\mathcal {P}}\) estimation. The special attention is paid to \(M_{\mathcal {P}}\) estimation that includes the asymmetry and the excess kurtosis, which are basic anomalies of empiric distributions of errors of geodetic or astrometric observations (in comparison to the Gaussian errors). It is assumed that the influence function of \(M_{\mathcal {P}}\) estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments \(\mu _{k},\, k=2,3,4\) , are the parameters of that system and thus, they are also the parameters of the chosen influence function. The \(M_{\mathcal {P}}\) estimation that includes the Pearson type IV and VII distributions ( \(M_{\mathrm{PD(l)}}\) method) is analyzed in great detail from a theoretical point of view as well as by applying numerical tests. The chosen distributions are leptokurtic with asymmetry which refers to the general characteristic of empirical distributions. Considering \(M\) -estimation with probabilistic models, the Gram–Charlier series are also applied to approximate the models in question ( \(M_{\mathrm{G-C}}\) method). The paper shows that \(M_{\mathcal {P}}\) estimation with the application of probabilistic models belongs to the class of robust estimations; \(M_{\mathrm{PD(l)}}\) method is especially effective in that case. It is suggested that even in the absence of significant anomalies the method in question should be regarded as robust against gross errors while its robustness is controlled by the pseudo-kurtosis.

This paper presents an estimation method allowing direct and robust outlier assessment of parameter shifts in functional models of geodetic observations. The systems adopted in the discussion refer to functional models used in geodetic network deformation analysis. The proposed method is a robustness-oriented development of Shift-Msplit estimation. The problem of robustness is solved using an additional random variable with realizations that are outliers. The paper shows that the application of this variable also facilitates the selection of the appropriate number of competitive models in Msplit(q) estimation. Two numerical examples explain the manner of Shift-Msplit∗ estimation performance and indicate the basic properties of the determined estimates.