On the DIA-PDF of the Huber-, Danish- and IGG-Estimator

Robust estimation refers to statistical methods used to estimate parameters in a way that is unduly influenced by violations of model assumptions. Different classes of robust estimators exist, such as the class of DIA-estimators and the class of robust M-estimators. DIA-estimators are designed to be robust against specified sets of mean-shift violations of the model, whereas M-estimators aim to reduce the impact of extreme observational errors by down-weighting their influence on the overall estimation. M-estimators are defined by their objective function, or its derivative, the influence function. The functions most often used in geodesy are those of Huber (Huber 1992), Danish-method (Krarup et al. 1980) and IGG-method (Yang 1999). An often voiced criticism of the M-class is that their properties of robustness are somewhat opaque, that the settings of their methods/functions and tuning parameters can be subjective and require careful considerations, and that, apart from their asymptotic properties, it is a challenge to obtain distributional results of their estimators. In this contribution we will resolve some of this criticism by showing how the distributional properties of the DIA-class (Teunissen 2018, 2024) can also be used to reveal those of the M-estimators. This will be demonstrated and analyzed for the Huber-, Danish- and IGG-method. As a result, an insightful diagnostic methodology is provided by means of which the workings of these robust estimators can be compared.