MM-Estimation for Robust Regression
Also known as: MM-estimation, MM robust regression, high-breakdown high-efficiency estimator, MM-Tahmin Edici
The MM-estimator is a robust linear regression method introduced by Victor J. Yohai in 1987. It combines the high breakdown point of an S-estimator with the high efficiency of an M-estimator, so it resists outliers strongly while still using the data efficiently when errors are well-behaved.
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When to use it
Use MM-estimation when you want linear regression coefficients that stay reliable in the presence of outliers in the response. It suits continuous outcomes with an approximately linear model and a reasonable sample (at least about 30 observations), and it tolerates a contamination fraction up to roughly 50%. It is the method of choice when you suspect outliers but still want efficiency close to OLS on the clean part of the data. It is less appropriate for very small samples, where convergence guarantees weaken, or when high-leverage points in the predictors require the initial S-stage to be tuned carefully.
Strengths & limitations
- Combines a high breakdown point (tolerates up to about 50% outliers) with high statistical efficiency close to OLS on clean data.
- Bounded redescending loss means extreme residuals lose influence rather than dominating the fit.
- Produces interpretable linear-regression coefficients while remaining far more resistant to contamination than least squares.
- Convergence guarantees weaken in very small samples (n < 20), where a simpler robust method may be safer.
- High-leverage points in the predictor space require careful tuning of the initial S-stage.
- Relies on the linear model being valid and is more computationally involved than ordinary least squares.
Frequently asked
How is MM-estimation different from ordinary least squares?
OLS minimises the sum of squared residuals, so a few extreme points can pull the line far off. MM-estimation replaces the squared loss with a bounded, redescending loss anchored to a robust scale, so large residuals stop influencing the fit. On clean data it behaves almost like OLS, but it stays reliable when up to half the data are contaminated.
What does 'high breakdown point and high efficiency' mean?
The breakdown point is the fraction of contaminated observations an estimator can tolerate before it gives an arbitrary answer; MM-estimation tolerates up to about 50%. Efficiency measures how well it uses clean data compared with the best possible estimator; MM-estimation can be tuned close to OLS efficiency. Yohai's contribution was achieving both at once.
Why does MM-estimation use two stages?
The first S-stage provides a robust estimate of the error scale with a high breakdown point but low efficiency. Fixing that scale, the second M-stage minimises a redescending loss to recover efficiency. The two stages together inherit the breakdown point of the first and the efficiency of the second.
When should I prefer a different robust method?
In very small samples (fewer than about 20 observations) the convergence of MM-estimation is less assured, so a median-based estimator such as Theil-Sen may be safer. When high-leverage points dominate the predictor space, a least-trimmed-squares fit can be more dependable.
Sources
- Yohai, V. J. (1987). High Breakdown-Point and High Efficiency Robust Estimates for Regression. Annals of Statistics, 15(2), 642-656. DOI: 10.1214/aos/1176350366 ↗
- Koller, M. & Stahel, W. A. (2011). Sharpening Wald-type Inference in Robust Regression for Small Samples. Computational Statistics & Data Analysis, 55(8), 2504-2515. DOI: 10.1016/j.csda.2011.02.014 ↗
How to cite this page
ScholarGate. (2026, June 1). MM-Estimation for Robust Regression. ScholarGate. https://scholargate.app/en/statistics/mm-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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