在“Foreground Estimation Based on Linear Regression Model with Fused Sparsity on Outliers”这篇文献里，在介绍最小二乘对噪声不鲁棒的时候提到了一个新的东西叫做M-estimators，好像就是之前所说的M估计，在大学里老师好像提过这个东西，但如今我却不知道这究竟是个什么东东了！经网上查阅，把相关东西记录在此，“以儆效尤”。

转载自：http://www.statisticalconsultants.co.nz/blog/m-estimators.html

M-estimators
Statistical Analysis Techniques, Robust Estimators, Alternatives to OLS
The three main classes of robust estimators are M,  L and  R.  Robust estimators are resistant to outliers and when used in  regression modelling, are robust to departures from the normality assumption. 
M-estimators are a maximum likelihood type estimator.  M estimation involves minimizing the following:
Where ρ is some function with the following properties: 
  • ρ(r) ≥ 0 for all r and has a minimum at 0
  • ρ(r) = ρ(-r) for all r
  • ρ(r) increases as r increases from 0, but doesn’t get too large as r increases
For  LAD:   ρ(r) = |r| 
For OLS:   ρ(r) = r 2 
Note that OLS doesn’t satisfy the third property, therefore it doesn’t count as a robust M-estimator. 
In the case of a linear model, the function to minimise will be:

Instead of minimising the function directly, it may be simpler to use the function’s first order conditions set to zero:

where:
If the ρ function can be differentiated, the M-estimator is said  to be a ψ-type. Otherwise, the M-estimator is said to be a ρ-type.

Lp subclass

L p is a subclass of M estimators.  An L p beta coefficient would be one that minimises the following:
Where 1≤ p ≤2.  
If p=1, it is the equivalent of  LAD and if p=2, it is the equivalent of OLS.
The lower p is, the more robust the L p will be to outliers.  The lower p is, the greater the number of iterations would be needed for the sum of |r| p to converge at the minimum.

Tukey’s bisquare M-estimator

Tukey proposed an M-estimator that has the following ρ(z i) function:
Where c is a constant and  , where s is the estimated scale parameter.

Tukey’s bisquare psi function leaves out any extreme outliers by giving them a zero weighting.

Huber's M-estimator

Huber proposed an M-estimator that has the following ρ(z i) function:
Where c is a constant and  , where s is the estimated scale parameter.

It essentially applies an LAD function to outliers and an OLS function to the other observations.

Andrews's M-estimator

Andrews (1974) proposed the following ρ(z i) function:
Where   , where s is the estimated scale parameter.

See also:

Regression 
LAD
L-estimators
R-estimators

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