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Robust estimation aims at developing point estimators that are not highly sensitive to errors in the data. However, the population parameters of interest are not identified under the assumptions of robust estimation, so the rationale for point estimation is not apparent. This paper shows that under the assumptions of robust estimation, population parameters can be bounded, even though they are not identified. Several features of the bounds are related to the breakdown point and gross-error sensitivity of robust estimation. A method for estimating the bounds is given and illustrated with an application to data on the distribution of household incomes in the U.S. It is argued that in the presence of errors in data, it is more natural to estimate the bounds than to attempt point estimation of unidentified parameters.


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