Least Median of Squares (LMS) Regression
Least Median of Squares Regression · Also known as: LMS, least median of squares regression, en küçük medyan kareler (LMS)
Least Median of Squares is a robust linear regression method introduced by Peter J. Rousseeuw in 1984. Instead of minimising the sum of squared residuals like ordinary least squares, it minimises the median of the squared residuals, which lets the fit resist contamination by up to roughly 50% outliers.
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When to use it
Use LMS for predicting or explaining a continuous outcome with a linear model when the data may contain a substantial share of outliers and you need a fit that is not pulled toward them. It assumes the linear model is valid and tolerates an outlier proportion of up to about 50%. A reasonable sample size (at least about 30 observations) is recommended; with very few observations its statistical efficiency drops sharply and a simpler median-based estimator is preferable.
Strengths & limitations
- Very high breakdown point: tolerates contamination of up to about 50% of the observations.
- Resists outliers far better than ordinary least squares, since extreme residuals do not affect the median.
- Requires no normality assumption on the errors.
- Low statistical efficiency compared with least squares when the data are clean.
- No closed-form solution; estimates come from a search over candidate fits, which is computationally heavier.
- Performs poorly in very small samples; with fewer than about 20 observations a simpler median-based method is preferable.
Frequently asked
How is LMS different from ordinary least squares?
OLS minimises the sum of squared residuals, so a few large errors dominate and outliers can dictate the fit. LMS minimises the median of the squared residuals, so the most extreme residuals are ignored and the line follows the bulk of the data.
What does the 50% breakdown point mean?
It means up to roughly half of the observations can be arbitrary outliers before the LMS fit can be forced to take an arbitrary value. This is the highest breakdown point an estimator can achieve, which is what makes LMS exceptionally robust.
Why is LMS less efficient than least squares?
Relying on the median rather than all residuals throws away information when the data are clean, so on outlier-free, well-behaved data LMS estimates are noisier than the least-squares estimates. The trade is robustness for efficiency.
What should I use when the sample is very small?
With fewer than about 20 observations LMS has very low efficiency, and below about 10 observations it is not meaningful. In those cases a simpler median-based robust estimator such as the Theil-Sen estimator is preferable.
Sources
- Rousseeuw, P. J. (1984). Least Median of Squares Regression. Journal of the American Statistical Association, 79(388), 871-880. DOI: 10.1080/01621459.1984.10477105 ↗
- Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley. ISBN: 978-0471735779
How to cite this page
ScholarGate. (2026, June 1). Least Median of Squares Regression. ScholarGate. https://scholargate.app/en/statistics/least-median-squares
Which method?
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