S-Estimator for Robust Regression
Also known as: S-estimation, robust S-regression, S-Tahmin Edici
The S-estimator is a robust linear-regression method, introduced by Rousseeuw and Yohai in 1984, that estimates the coefficients by minimising a robust M-estimate of the residual scale rather than the variance of the residuals. By driving down a bounded measure of residual spread it can attain a breakdown point of up to 50%, so it stays reliable even when a large share of the data are outliers, and it provides the first stage of the well-known MM-estimator.
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When to use it
Use the S-estimator when you want a linear regression on a continuous outcome but suspect that a substantial fraction of the observations may be outliers, including in the response, and you need an estimator whose breakdown point stays near 50%. It assumes the linear model is correct for the clean majority of the data and works best with at least about 30 observations. It is well suited to contaminated cross-sectional or longitudinal data, but it is less efficient than least squares when the errors are actually clean and normal, and its efficiency drops in the presence of high-leverage points, where an MM-estimator is preferable.
Strengths & limitations
- High breakdown point of up to 50%, so it tolerates a large proportion of outliers in the data.
- Minimises a robust, bounded measure of residual scale, so extreme residuals cannot dominate the fit.
- Serves as the high-breakdown first stage of the MM-estimator, combining robustness with a foundation for higher efficiency.
- On its own the S-estimator has comparatively low Gaussian efficiency, which is why it is often refined into an MM-estimator.
- It can struggle to converge in very small samples (n below about 20), where a simpler robust method is safer.
- Its efficiency drops when high-leverage points are present; an MM-estimator is then preferable.
Frequently asked
How does the S-estimator differ from ordinary least squares?
Least squares minimises the sum of squared residuals, so a few extreme points can pull the line toward them. The S-estimator minimises a robust, bounded estimate of residual scale instead, which caps the influence of outliers and lets it tolerate up to 50% contamination.
What is the breakdown point and what value does the S-estimator achieve?
The breakdown point is the largest fraction of contaminated data an estimator can withstand before giving an arbitrarily wrong answer. By choosing the constant K in its scale equation appropriately, the S-estimator can reach a breakdown point of up to 50%.
Why is the S-estimator described as the first stage of the MM-estimator?
An MM-estimator first computes a high-breakdown S-estimate of the regression and its residual scale, then refines the coefficients with an M-step to gain Gaussian efficiency while keeping the robustness. The S-estimate supplies that robust starting point.
When should I prefer a different method?
In very small samples (n below about 20) convergence can be unreliable, so a simpler robust fit such as the Theil-Sen estimator is safer. When high-leverage points are present the S-estimator loses efficiency, and an MM-estimator is the better choice.
Sources
- Rousseeuw, P. J. & Yohai, V. J. (1984). Robust Regression by Means of S-Estimators. In Robust and Nonlinear Time Series Analysis (Lecture Notes in Statistics, Vol. 26, pp. 256-272). Springer. DOI: 10.1007/978-1-4615-7821-5_15 ↗
- Maronna, R. A., Martin, R. D., Yohai, V. J. & Salibián-Barrera, M. (2019). Robust Statistics: Theory and Methods (with R) (2nd ed.). Wiley. ISBN: 978-1119214687
How to cite this page
ScholarGate. (2026, June 1). S-Estimator for Robust Regression. ScholarGate. https://scholargate.app/en/statistics/s-estimator
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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