Robust Statistics - M-estimators

M-estimators

(The mathematical context of this paragraph is given in the section on empirical influence functions.)

Historically, several approaches to robust estimation were proposed, including R-estimators and L-estimators. However, M-estimators now appear to dominate the field as a result of their generality, high breakdown point, and their efficiency. See Huber (1981).

M-estimators are a generalization of maximum likelihood estimators (MLEs). What we try to do with MLE's is to maximize or, equivalently, minimize . In 1964, Huber proposed to generalize this to the minimization of, where is some function. MLE are therefore a special case of M-estimators (hence the name: "Maximum likelihood type" estimators).

Minimizing can often be done by differentiating and solving, where (if has a derivative).

Several choices of and have been proposed. The two figures below show four functions and their corresponding functions.

For squared errors, increases at an accelerating rate, whilst for absolute errors, it increases at a constant rate. When Winsorizing is used, a mixture of these two effects is introduced: for small values of x, increases at the squared rate, but once the chosen threshold is reached (1.5 in this example), the rate of increase becomes constant. This Winsorised estimator is also known as the Huber loss function.

Tukey's biweight (also known as bisquare) function behaves in a similar way to the squared error function at first, but for larger errors, the function tapers off.