Theil-Sen Estimator
Also known as: Theil-Sen Tahmincisi, Theil-Sen regression, median slope estimator, Sen's slope estimator, Kendall-Theil robust line
The Theil-Sen estimator is a robust linear regression method that estimates the slope as the median of the slopes computed over all pairs of data points. Introduced by Henri Theil in 1950 and extended by P. K. Sen in 1968, it tolerates outliers in the response with a breakdown point of about 29%.
Key highlights
- Highly robust to outliers in the response, with a breakdown point of about 29%.
- Distribution-free: it makes no normality assumption about the errors.
- Conceptually simple and tied to Kendall's tau, giving an interpretable, rank-based slope.
Intuition
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How it works
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When to use it
Use Theil-Sen when you have a continuous outcome and one (or very few) continuous predictors, the relationship is approximately linear, and you want a slope estimate that is not distorted by outliers or heavy-tailed errors. It does not require normally distributed residuals and works well from about 20 observations upward. It is a strong alternative to ordinary least squares when a small fraction of the data is contaminated, but it is less suited to very small samples or data dominated by high-leverage points.
Strengths & limitations
- Highly robust to outliers in the response, with a breakdown point of about 29%.
- Distribution-free: it makes no normality assumption about the errors.
- Conceptually simple and tied to Kendall's tau, giving an interpretable, rank-based slope.
- Primarily designed for a single (or very few) predictors, so it does not scale to high-dimensional multiple regression like OLS.
- In very small samples (n < 10) the median slope estimate becomes unreliable.
- High-leverage points in the predictor can still exceed the single-variable breakdown point and distort the fit.
Common pitfalls
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Applications
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Frequently asked
What is the breakdown point of Theil-Sen?
About 29%, meaning roughly up to a quarter of the response values can be arbitrarily corrupted before the slope estimate breaks down. This is far more robust than ordinary least squares, whose breakdown point is effectively zero.
How does Theil-Sen relate to Kendall's tau?
Sen's 1968 formulation links the median-of-slopes estimator to Kendall's rank correlation tau, so the slope is essentially a rank-based, distribution-free quantity. This is why it makes no normality assumption.
Can I use Theil-Sen with several predictors?
It is designed for a single or very few predictors. Its pairwise-slope construction does not extend cleanly to high-dimensional multiple regression, so for many predictors a different robust or penalised method is more appropriate.
When should I prefer another method?
For very small samples (n < 10) a permutation test is more reliable, and when high-leverage points are present least trimmed squares better preserves robustness than Theil-Sen.
Sources
- 1.Sen, P. K. (1968). Estimates of the Regression Coefficient Based on Kendall's Tau. Journal of the American Statistical Association, 63(324), 1379-1389.
- 2.Theil, H. (1950). A Rank-Invariant Method of Linear and Polynomial Regression Analysis. Proceedings of the Royal Netherlands Academy of Sciences, 53, 386-392, 521-525, 1397-1412.
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ScholarGate. (2026, June 1). Theil-Sen Estimator. ScholarGate. https://scholargate.app/statistics/theil-sen-estimator