M-estimation with probabilistic models of geodetic observations

Abstract

The paper concerns M-estimation with probabilistic models of geodetic observations that is called M P estimation. The special attention is paid to M 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 P estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments μ 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 P estimation that includes the Pearson type IV and VII distributions (M 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 G−C method). The paper shows that M P estimation with the application of probabilistic models belongs to the class of robust estimations; M 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.

MP estimation applied to platykurtic sets of geodetic observations

“…(11). Wiśniewski (2014) proposed the name M P estimates for all the solutions of Eq. (7) which are based on the infl uence functions ψ(v i )  IF(v i , F X ) and the explicit families of distributions P = {P θ i : θ i  Θ}.…”

“…In the case of asymmetric distribution, the mode M 0 does not coincide with the expected value E(v). Considering such a requirement, Wiśniewski (2014) proposed to modify the differential equation of PD-system in the following way (13) where s = M 0 -E(v) = σc 1 /(c 0 + 3c 2 ). Taking in account the properties of the infl uence function ψ(v) = -Ω(v) in the context of M P estimation, it is worth considering two versions of that function, namely (14) wherein, if β 2 = 3 then ψ lep (v) = ψ mez (v), and ψ mez (v) is the infl uence function given for mesokurtic distributions (lep -leptokurtic distributions, plt -platykurtic distributions).…”

“…M P estimation with such a weight function was discussed in detail in (Wiśniewski, 2014). Such a property, which is advisable from the practical point of view, does not apply to the infl uence functions in the case of platikurtic distributions.…”

“…The general properties of the Pearson distributions are discussed by Elderton (1953) or Friori and Zenga (2009). The several selected distributions of that system were applied in astronomy (Dzhun', 1992(Dzhun', , 2012 and in geodesy (Wiśniewski, 1987(Wiśniewski, , 2014.…”

“…Note that the infl uence function is based on distribution functions for whole PD-system. Considering such assumptions and the main idea proposed by Dzun'a (2011), Wiśniewski (2014 brought and analyzed a new solution called M P estimation. The paper in question focused on such variants of the method which are referred to mesokurtic or leptokurtic distributions.…”