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 $\rho$, which satisfy

• R1 – $\rho$ is symmetric, continuously differentiable and $\rho (0)=0$.
• R2 – there exists $c>0$ such that $\rho$ is strictly increasing on $[c,\infty ]$

For any sample $\{r_{1},...,r_{n}\}$ of real numbers, we define the scale estimate $s(r_{1},...,r_{n})$ as the solution of

${\textstyle {\frac {1}{n}}\sum _{i=1}^{n}\rho (r_{i}/s)=K}$,

where $K$ is the expectation value of $\rho$ 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 $\theta$, we obtain residuals $s(r_{1}(\theta ),...,r_{n}(\theta ))$ by solving the equation of scale above, where $\rho$ satisfy R1 and R2. The S-estimator ${\hat {\theta }}$ is defined by

${\hat {\theta }}=\min _{\theta }\,s(r_{1}(\theta ),...,r_{n}(\theta ))$

and the final scale estimator ${\hat {\sigma }}$ is then

${\hat {\sigma }}=s(r_{1}({\hat {\theta }}),...,r_{n}({\hat {\theta }}))$.