Testing normality of chosen R-estimates used in deformation analysis

Abstract The normal distribution is one of the most important distribution in statistics. In the context of geodetic observation analyses, such importance follows Hagen’s hypothesis of elementary errors; however, some papers point to some leptokurtic tendencies in geodetic observation sets. In the case of linear estimators, the normality is guaranteed by normality of the independent observations. The situation is more complex if estimates and/or the functional model are not linear. Then the normality of such estimates can be tested theoretically or empirically by applying one of goodness-of-fit tests. This paper focuses on testing normality of selected variants of the Hodges-Lehmann estimators (HLE). Under some general assumptions the simplest HLEs have asymptotical normality. However, this does not apply to the Hodges-Lehmann weighted estimators (HLWE), which are more applicable in deformation analysis. Thus, the paper presents tests for normality of HLEs and HLWEs. The analyses, which are based on Monte Carlo method and the Jarque–Bera test, prove normality of HLEs. HLWEs do not follow the normal distribution when the functional model is not linear, and the accuracy of observation is relatively low. However, this fact seems not important from the practical point of view.