Accuracy of the Hodges–Lehmann estimates computed by applying Monte Carlo simulations

Abstract

The paper concerns assessing the accuracy of some variants of robust R-estimates, namely the Hodges-Lehmann estimates, which can be applied, for example, in deformation analyses. Such estimates are robust against outlying observations and in some cases they are a good alternative for more conventional methods of estimation, for example, in testing stability of the potential reference points. Considering such an application, or in general estimation of displacements of network points, one should of course know accuracy of the estimators. Since R-estimates are based on ranks it is not obvious how to compute their accuracy (the law of variance propagation cannot be applied here). This paper presents one of the possible approaches, namely application of Monte Carlo simulations. If we make certain assumptions concerning the distribution of observation errors, we can assess the accuracy of chosen R-estimates. Usually, we assume that the observation errors are normally distributed, however, we can also consider some distributions with positive or negative kurtosis, and in the latter case we may apply the system of Johnson's distributions to simulate the observations. In the paper, the accuracy of R-estimates was computed in relation to the accuracy of LS estimates, which is advisable from a practical point of view. It turned out that the accuracy of R-estimates is a little bit worse than the accuracy of LS estimates in most of the cases. However, there are also some cases when R-estimates are more accurate, for example, for leptokurtic distributions of the observations. An example application of R-estimates in deformation analysis was also presented.

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

“…The general assumptions of the basic HLEs lead to the conclusion that all sample elements (observations) must have the same accuracy [18,116]. However, in geodetics, the observations usually have different accuracy.…”

“…For other models, one should instead focus on estimators of the variancecovariance matrices of the robust parameter estimates. Since all robust estimators discussed here are not linear, providing strict forms of such matrices in rigorous theory in such a context is not an easy task [105,116,158]. Therefore, in many practical applications, one should calculate the necessary assessment using simulation methods, usually Monte Carlo procedures, with all the pros and cons of such an approach [116,159].…”

Robust Estimation of Deformation from Observation Differences Using Some Evolutionary Optimisation Algorithms

“…On the other hand, one of the most important tasks in the deformation analysis of geodetic networks is the selection of the optimal datum solution for the parameters of geodetic networks. Among deformation analysis methods that have been researched for theoretical and practical applications, it is important to mention the methods of conventional deformation analysis (CDA) [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 ] and methods of robust deformation analysis (RDA) based on M estimation [ 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ], M split estimation [ 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ] and R estimation [ 33 , 34 , 35 ]. In conventional deformation analysis, it is essential to identify stable datum points, i.e., potential reference points (PRPs) of the network stabilised outside the zone of expected deformations.…”

Efficacy of Msplit Estimation in Displacement Analysis

“…The least squares method (LS-method) is still the most popular approach in such an analysis, note that LS-estimates are often supplemented with respective statistical tests (e.g., [4,5,6]). However, some unconventional methods are also in use, for example, robust M-estimation [7,8] or R-estimation [9,10,11,12,13,14]. In the case of relative networks, one can also apply methods of free adjustment (e.g., [15,16,17,18]).…”