There are several robust regression methods like LAR-(aka LAV-, LAD-, L1-Norm-)Regression, Quantil-Regression, M-Estimator, ... They are assumed to be especially appropriate for data, that does not fulfill the 5 OLS conditions.

The major part of the robust regression literature (I read) argues abstractly with the breakdownpoint which robust estimator should generally be preferred.

The other part of the robust regression literature (I read) argues, the best robust estimator depends from the next best comparable theoretical distribution. E.g. will the LAR-estimator most probably be the best robust estimator at approximate Laplace distribution (although it has a worse breakdown point than quantile-estimator/ M-estimator).

Question:

How do I choose the best robust regression model from multiple robust estimators for data, that does (graphically obviously) not fullfill the OLS-conditions? ]]>