What is "s-estimator"

The goal of S-estimators is to have a simple high-breakdown regression estimator, which share the flexibility and nice asymptotic properties of M-estimators. The name "S-estimators" was chosen as they are based on estimators of scale.

We will consider estimators of scale defined by a function ρ, which satisfy

R1 - ρ is symmetri, continuously differentiable and ρ(0) = 0.

R2 - there exists c > 0 such that ρ is strictly increasing on [c, ∞[

For any sample {r, ..., r} of real numbers, we define the scale estimate s(r, ..., r) as the solution of

$\frac{1}{n}\sum_{i=1}^n \rho(r_i/s) = K$,

where K is the expectation value of ρ for a standard normal distribution. (If there are more solutions to the above equation, then we take the one with the smallest solution for s; if there is no solution, then we put s(r, ..., r) = 0 .)

Definition:

Let (x, y), ..., (x, y) be a sample of regression data with p-dimensional x. For each vector θ, we obtain residuals s(r(θ), ..., r(θ)) by solving the equation of scale above, where ρ satisfy R1 and R2. The S-estimator θ is defined by

minimize s(r(θ), ..., r(θ))

and the final scale estimator is

θ = s(r(θ), ..., r(θ)) .