Which statement about M-estimators is correct?

• A. They are always derived from the mean.
• B. They are always parametric.
• C. They are the same as the trimmed mean.
• D. Unlike a trimmed mean, the amount of trimming used to remove outliers is determined empirically.

Answer: (D)

M-estimators use a robust loss function to downweight the influence of outliers rather than simply trimming data. They aren’t tied to the mean as the sole basis, and they aren’t just the trimmed mean; instead, they assign weights or penalties to residuals through a chosen ρ or ψ function. The amount of downweighting is governed by a tuning constant in that function, which is typically selected empirically to achieve a desired balance between robustness and efficiency. That means the “extent” of outlier handling is not a fixed, pre-specified trim amount but is determined by the data and the tuning choice. For example, a Hub­er M-estimator uses a threshold that decides how aggressively large residuals are downweighted, and that threshold is often chosen via empirical methods or simulations. That’s why this statement best captures how M-estimators operate.