Definition
In 1964, Peter Huber proposed generalizing maximum likelihood estimation to the minimization of
where ρ is a function with certain properties (see below). The solutions
are called M-estimators ("M" for "maximum likelihood-type" (Huber, 1981, page 43)); other types of robust estimator include L-estimators, R-estimators and S-estimators. Maximum likelihood estimators are thus a special case of M-estimators. With suitable rescaling, M-estimators are special cases of extremum estimators (in which more general functions of the observations can be used).
The function ρ, or its derivative, ψ, can be chosen in such a way to provide the estimator desirable properties (in terms of bias and efficiency) when the data are truly from the assumed distribution, and 'not bad' behaviour when the data are generated from a model that is, in some sense, close to the assumed distribution.
Read more about this topic: M-estimator
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