MP estimation applied to platykurtic sets of geodetic observations

Abstract

M P estimation is a method which concerns estimating of the location parameters when the probabilistic models of observations differ from the normal distributions in the kurtosis or asymmetry. The system of Pearson's distributions is the probabilistic basis for the method. So far, such a method was applied and analyzed mostly for leptokurtic or mesokurtic distributions (Pearson's distributions of types IV or VII), which predominate practical cases. The analyses of geodetic or astronomical observations show that we may also deal with sets which have moderate asymmetry or small negative excess kurtosis. Asymmetry might result from the infl uence of many small systematic errors, which were not eliminated during preprocessing of data. The excess kurtosis can be related with bigger or smaller (in relations to the Hagen hypothesis) frequency of occurrence of the elementary errors which are close to zero. Considering that fact, this paper focuses on the estimation with application of the Pearson platykurtic distributions of types I or II. The paper presents the solution of the corresponding optimization problem and its basic properties. Although platykurtic distributions are rare in practice, it was an interesting issue to fi nd out what results can be provided by M P estimation in the case of such observation distributions. The numerical tests which are presented in the paper are rather limited; however, they allow us to draw some general conclusions.

Accuracy Characteristics of the Selected Diagnostics Methods and the Adjustment of Geodetic Observations

“…This modification assumes, among other things, a change of weights for individual observations. It is an iterative method, requiring several repetitions until the result meets the adopted criteria [28].…”

“…Their favourable properties result from the modified objective function and its derivative, i.e. the influence function and associated weighting function [28,29].…”

Robust procedures in processing measurements in geodesy and surveying: a review

“…One can, of course, list some other methods that belong to (or might be considered as a development of) the Mestimation class and which can be applied to geodetic problems, e.g. Welsch method, German-McClure method, the principle of the alternative choice, the method of growing rigor, the least median of squares, or the least trimmed squares, sign-constrained robust least squares method, the alternative to LAD (ALAD), M P estimation [7,76,[83][84][85][86][87].…”

Testing normality of chosen R-estimates used in deformation analysis

“…Of course, one can choose other test which can be more powerful (for example, Thadewald and Büning, 2007); however, JB test is very simple in use and for the purpose of this paper application of this test seems enough. JB test is based on the following test statistic (Jarque and Bera, 1980;Wiśniewski, 2017)…”