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 , which satisfy

  • R1 – is symmetric, continuously differentiable and .
  • R2 – there exists such that is strictly increasing on

For any sample of real numbers, we define the scale estimate as the solution of

,

where is the expectation value of 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 , we obtain residuals by solving the equation of scale above, where satisfy R1 and R2. The S-estimator is defined by

and the final scale estimator is then

.[1]

References

  1. 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
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