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.
Highlights
Introduction and motivationRobust methods of estimation are well known and applied in several geodetic or surveying problems
The law of propagation of variance is the most often method applied to assess such accuracy; such an approach is impossible for some estimation methods
A good alternative is the method that is based on Monte Carlo simulations
Summary
Robust methods of estimation are well known and applied in several geodetic or surveying problems. There are two main variants of HLWE (or HLE, respectively), namely R-estimates of the expected value and R-estimates of the shift between two sets of observations The latter kind of R-estimates is especially useful in deformation analyses (see, Duchnowski 2010, 2013). In the case of deformation analyses, where the samples contains usually a few elements, more robust are R-estimates of the shift, they are more important from the practical point of view (Duchnowski 2011, 2013) As it was already mentioned, the law of propagation of variance cannot be used to compute the variance of HLWE (or HLE) directly from the variances of the observations applied. In the case of shift estimates, we assume that n = m, namely both simulated samples have the same number of elements, which is acceptable from the practical point of view, for example, when we consider deformation analyses. The accuracy of R-estimates will be compared to accuracy of respective LSEs, namely an arithmetic mean or a weighted arithmetic mean (estimation of the expected value), and the respective differences of arithmetic or weighted arithmetic means (in the case of the shift estimates)
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