Robust estimation via minimum distance methods

Summary Given a fixed parametric family {P } it is desired to estimate. However, due to contaminations of various sorts, the data actually collected by the statistician follow a distribution that is close to, but possibly distinct from, theP 's. It is proved that, under these conditions and in an appropriate asymptotic framework, the minimum distance estimators of the Cramér-von Mises type are robust. Specification of a Cramér-von Mises weight functionH defines a notion of distance; each such choice ofH then delineates the kind of contamination possible, and leads to an estimator which defends optimally against it. When the theory is specialized to location models, various choices ofH lead to estimators asymptotically equivalent to such familiar ones as trimmed mean, median, Hodges-Lehmann estimator, and so forth. The framework developed herein provides some guidelines for choosing among the possible estimators, and suggests that the standard Cramér-von Mises estimator of location is probably as good a robust estimator as any.Supported by NSF grant MCS 75-10376