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

,

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

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