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 $[c,\mathrm{\infty }]$

For any sample $\{r_1,...,r_n\}$ of real numbers, we define the scale estimate $s(r_1,...,r_n)$ as the solution of

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

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

$\hat{\theta }=\underset{\theta }{\min }s(r_1(\theta ),...,r_n(\theta ))$

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

$\hat{\sigma }=s(r_1(\hat{\theta }),...,r_n(\hat{\theta }))$.