Which data condition makes robust least squares most beneficial?

• A. Data with no outliers.
• B. Data with occasional outliers due to measurement errors.
• C. Data with purely random Gaussian noise.
• D. Data with all measurements identical.

Answer: (B)

Robust least squares shines when you have outliers that don’t belong to the underlying pattern but can heavily skew the fit if treated like ordinary data. It uses a loss function that down-weights large residuals, so those few abnormal points don’t pull the line or plane off course. That makes the estimated relationship reflect the central trend of most measurements rather than being dominated by a handful of measurement errors.

If there aren’t outliers, ordinary least squares is typically more efficient and provides the best linear unbiased estimates under the usual noise assumptions, so robust methods don’t offer a real advantage. When the noise is purely Gaussian, OLS achieves minimum variance among unbiased estimators, whereas robust methods may introduce a bit of bias or inefficiency. If all measurements are identical, there’s no information to fit, so neither method helps, and the problem is degenerate.

So the scenario where robust least squares is most beneficial is when there are occasional outliers due to measurement errors.