Statistical Consultants Ltd

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:

sum rho

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) = r²

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:

sum rho regression

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.

L_p subclass

L_pis a subclass of M estimators. An L_pbeta coefficient would be one that minimises the following:

Lp rho function

Where 1≤ p ≤2.
If p=1, it is the equivalent of LAD and if p=2, it is the equivalent of OLS.

Lp comparison graph

The lower p is, the more robust the L_pwill 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.