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 symmetric, continuously differentiable and ρ ( 0 ) = 0 .
R2 - there exists c > 0 such that ρ is strictly increasing on [ c , ∞ [
For any sample { r 1 , . . . , r n } of real numbers, we define the scale estimate s ( r 1 , . . . , r n ) as the solution of
1 n ∑ i = 1 n ρ ( 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 1 , . . . , r n ) = 0 .)
Definition:
Let ( x 1 , y 1 ) , . . . , ( x n , y n ) be a sample of regression data with p-dimensional x i . For each vector θ , we obtain residuals s ( r 1 ( θ ) , . . . , r n ( θ ) ) by solving the equation of scale above, where ρ satisfy R1 and R2. The S-estimator θ is defined by
minimize s ( r 1 ( θ ) , . . . , r n ( θ ) )
and the final scale estimator is
θ = s ( r 1 ( θ ) , . . . , r n ( θ ) ) .
