Redescending M-estimator

Redescending M-estimators are very popular Ψ-type M-estimators which have Ψ functions that are non-decreasing near the origin, but decreasing toward 0 far from the origin.

Their ψ functions can be chosen to redescend smoothly to zero, so that they usually satisfy Ψ(x) = 0 for all x with |x| > r, where r is referred to as the minimum rejected point.

Due to these properties of the ψ function, these kinds of estimators are very efficient, have a high breakdown point, and, unlike other outlier rejection techniques, they do not suffer from a masking effect.

Their efficiency resultant off the fact that they completely reject gross outliers, and do not completely ignore moderately large outliers (like median).

Advantages

Redescending M-estimators have very low breakdown points (close to 0.5), and their Ψ function can be chosen to redescend smoothly to 0.This means that moderately large outliers are not ignored completely, and is greatly improves the efficiency of the redescending M-estimator.

The redescending M estimators are slightly more efficient than the Huber estimator for several symmetric, wider tailed distributions, but much more efficient (about 20% more) than the Huber estimator for the Cauchy distribution. This is because they completely reject gross outliers, while the Huber estimator effectively treats them the same as moderate outliers.

Unlike other outlier rejection techniques, they do not suffer from masking effects.

Disadvantages

Redescending estimators are usually do not have unique solution to the M-estimating equation.

Choosing redescending Ψ functions

When choosing a redescending Ψ functions we must take care that it does not descend too steeply, which may have a very bad influence on the denominator in the expression for the asymptotic variance

: $frac\{int\; Psi^2\; ,\; dF\; ,!\}\{(int\; Psi\text{'}\; ,\; dF\; ,!)^2\}$

where "F" is the mixture model distribution.

This effect is particularly harmful when a large negative values of ψ'("x") combines with a large positive values of ψ²("x"), and there is a cluster of outliers near "x".

Examples

1. Hampel's three-part M estimators have Ψ functions which are odd functions and defined for any x by:

::

This function is plotted in the following figure for a=1.645, b=3 and r=6.5.

2. Tukey's biweight or bisquare M estimators have Ψ functions for any positive k, which defined by:

:$Psi(x)=x(1-(x/k)^2)^2\; ;\; |x|le\; k$

This function is plotted in the following figure for k=5.

3. Andrew's sine wave M estimator has the following Ψ function:

:$Psi(x)=sin\{(x)\};\; -pi\; le\; x\; lepi$

This function is plotted in the following figure.

References

* "Robust Estimation and Testing", Robert G. Staudte and Simon J. Sheather, Wiley 1990.
* "Robust Statistics",Huber, P., New York: Wiley, 1981.

ee also

*M-estimator
*Robust statistics