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 $\backslash rho$, which satisfy

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

For any sample $\backslash \{r\_1,\; ...,\; r\_n\backslash \}$ of real numbers, we define the scale estimate $s(r\_1,\; ...,\; r\_n)$ as the solution of

$\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^n\; \backslash rho(r\_i/s)\; =\; K$,

where $K$ is the expectation value of $\backslash rho$

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 $\backslash theta$

, we obtain residuals $s(r\_1(\backslash theta),...,\; r\_n(\backslash theta))$ by solving the equation of scale above, where $\backslash rho$ satisfy R1 and R2. The S-estimator $\backslash hat\backslash theta$ is defined by

$\backslash hat\backslash theta\; =\; \backslash min\_\backslash theta\; \backslash ,\; s(r\_1(\backslash theta),...,\; r\_n(\backslash theta))$

and the final scale estimator $\backslash hat\; \backslash sigma$ is then

$\backslash hat\backslash sigma\; =\; s(r\_1(\backslash hat\backslash theta),\; ...,\; r\_n(\backslash hat\backslash theta))$.P. Rousseeuw and V. Yohai, Robust Regression by Means of S-estimators, from the book: Robust and nonlinear time series analysis, pages 256–272, 1984