Robustness properties of quasi-linear means with application to the Laspeyres and Paasche indices

Li, Fang & Tian (1994) assert that special quasi-linear means should be preferred to the simple arithmetic mean for robustness properties. The strategy that is used to show robustness is completely detached from the concepts wellknown from the theory of robust statistics. Robustness of estimators can be verified with tools from robust statistics, e.g. the influence function or the breakdown point. On the other hand it seems that robust statistics is not interested in quasi-linear means. Therefore, we compute influence functions and breakdown points for quasi-linear means and show that these means are not robust in the sense of robust statistics if the generator is unbounded. As special cases we consider the Laspeyres, the Paasche and the Fisher indices.