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 For any sample
of real numbers, we define the scale estimate
as the solution of
ρ (
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
Definition: Let
be a sample of regression data with p-dimensional
For each vector
θ
, we obtain residuals
( θ ) , .
( θ ) )
by solving the equation of scale above, where
ρ
satisfy R1 and R2.
The S-estimator
is defined by
min
and the final scale estimator
σ ^
σ ^