Tau (τ) Estimator of Regression
Also known as: tau regression estimator, robust tau regression, Tau-Tahmin Edici
The Tau estimator is a robust linear regression method introduced by Yohai and Zamar in 1988 that fits the model by minimising an efficient τ-scale of the residuals. It builds on the scale estimate of the S-estimator to combine a high breakdown point with high statistical efficiency, and is often used as an alternative to the MM-estimator in small samples.
Key highlights
- Combines a high breakdown point (resistance to outliers) with high statistical efficiency.
- A practical robust alternative to the MM-estimator, especially in small-to-moderate samples.
- Does not require normally distributed errors, so it tolerates contaminated and heavy-tailed data.
Intuition
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How it works
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When to use it
Use the Tau estimator for continuous outcomes modelled by a linear regression when you suspect outliers or contamination and need robust coefficients without sacrificing much efficiency. It assumes the linear model form is valid and is intended for small-to-moderate sample sizes (roughly 30 or more observations), where it serves as an alternative to the MM-estimator. It is less suitable when the sample is very small or when strong leverage points dominate the design.
Strengths & limitations
- Combines a high breakdown point (resistance to outliers) with high statistical efficiency.
- A practical robust alternative to the MM-estimator, especially in small-to-moderate samples.
- Does not require normally distributed errors, so it tolerates contaminated and heavy-tailed data.
- With very small samples (n < 20) the convergence guarantee weakens.
- Leverage points can still affect the underlying S-stage of the estimator.
- Computationally heavier than ordinary least squares and typically relies on iterative robust-regression routines.
Common pitfalls
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Applications
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Frequently asked
How does the Tau estimator differ from the MM-estimator?
Both target high breakdown and high efficiency, but the Tau estimator minimises an efficient τ-scale of the residuals built on an S-scale, while the MM-estimator follows an S-stage with an M-estimation step. In small-to-moderate samples the Tau estimator is often used as an alternative to the MM-estimator.
What is a breakdown point and why does it matter here?
The breakdown point is the fraction of contaminated observations an estimator can tolerate before producing arbitrarily wrong results. The Tau estimator achieves a high breakdown point, so a substantial share of outliers will not derail the fit.
When should I avoid the Tau estimator?
Avoid it with very small samples (n < 20), where its convergence guarantee weakens — a Theil-Sen estimator is more dependable. When strong leverage points dominate the design, the S-stage can be affected and an MM-estimator may be preferable.
Does the Tau estimator require normally distributed errors?
No. It does not assume normality and is designed to remain reliable under contamination and heavy-tailed errors, which is precisely why it is preferred over ordinary least squares in those settings.
Sources
- 1.Yohai, V. J., & Zamar, R. H. (1988). High Breakdown-Point Estimates of Regression by Means of the Minimization of an Efficient Scale. Journal of the American Statistical Association, 83(402), 406-413.
- 2.Maronna, R. A., & Zamar, R. H. (2002). Robust Estimates of Location and Dispersion for High-Dimensional Datasets. Technometrics, 44(4), 307-317.
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Cite this page
ScholarGate. (2026, June 1). Tau Estimator. ScholarGate. https://scholargate.app/statistics/tau-estimator