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.