M-estimation with probabilistic models of geodetic observations

Abstract

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.