An MM-estimator combines a high-breakdown regression start, robust scale and an efficient final M step. A fixed-scale bisquare example shows why a zero score alone is insufficient.
MM-estimator
An MM-estimator is a robust regression estimator built from a high-breakdown initial fit, a robust residual scale and a final M-estimation step tuned for efficiency. In a factor regression, it offers a way to stop a small contaminated subset from dominating the estimated exposures while retaining precision when the assumed clean-error model fits well.
The construction matters. Replacing squared residuals with a bounded loss and accepting any numerical solution does not establish the MM guarantees.
Protection first, efficiency second
Assume s0>0. Write ri(β)=yi−xi⊤β, including an intercept in xi when appropriate. Yohai's original construction, Section 2, proceeds in three stages:
Obtain a regression fit β~ with a high breakdown point. An S-estimator is a common starting choice.
Estimate scale s0 robustly from its residuals. For a bounded loss ρ0, the scale equation has the form
n1i=1∑nρ0(s0ri(β~))=b.
Hold s0 fixed and refine the coefficients using an efficient redescending score ψ1=ρ1′. Select a solution satisfying both
i=1∑nψ1(s0ri(β^))xi=0,Q1(β^)≤Q1(β~),
where Q1(β)=∑iρ1(ri(β)/s0).
The original conditions include continuous, even, bounded losses increasing with absolute residual before saturation, a common supremum a, ρ1≤ρ0, and b/a=1/2. The tuning and scale calibration must be compatible. A local solution can qualify when it satisfies the required conditions; an arbitrary score root cannot. The objective comparison is part of the definition, rather than an optional diagnostic.
MASS's implementation documentation gives a concrete calibrated example: an S start and fixed scale followed by Tukey's biweight, with tuning chosen for approximately 95% relative efficiency at normal errors. That is a model-based asymptotic comparison with least squares. It is neither a prediction-accuracy percentage nor a promise for a tiny contaminated sample. Its stated near-one-half breakdown interpretation assumes sample size large relative to the number of fitted coefficients.
A final step you can check by hand
Consider five synthetic observations, in arbitrary common units:
0,0,1,1,100.
This is an intercept-only illustration. Stipulate the initial location θ~=0.5 and fixed scale s0=2/3≈0.816497. These values are inputs for demonstrating the final M stage; they are not a Gaussian-calibrated MM fit.
Use the bisquare score with c=4.685:
ψc(u)={u(1−(u/c)2)2,0,∣u∣<c,∣u∣≥c.
At θ=0.5, the four central standardised residuals are two copies each of −3/8 and +3/8, approximately ±0.612372. Their scores are approximately −0.591627,−0.591627,+0.591627,+0.591627. The residual for 100 is about 121.862115, beyond the cutoff, and contributes zero. The total score is exactly zero by symmetry. By comparison, the ordinary sample mean is 20.4.
But θ=50 also gives a zero score: every standardised residual is outside the cutoff. To distinguish the two, use the normalised bisquare loss
ρc(u)={1−(1−(u/c)2)3,1,∣u∣<c,∣u∣≥c.
Its derivative is (6/c2)ψc, so it has the same score roots. Summing this loss gives Q(0.5)≈1.201536 and Q(50)=5. The remote plateau fails the objective comparison against the stipulated start. This demonstrates why the score equation alone is insufficient; it establishes no breakdown point or Gaussian efficiency for the toy procedure.
Synthetic final-M-stage example: with observations 0, 0, 1, 1, 100, location 0.5 and fixed scale square root of 2/3, bisquare scores are about minus 0.5916, minus 0.5916, plus 0.5916, plus 0.5916 and zero. Location 50 also has zero total score, but normalised loss 5 exceeds 1.2015 at 0.5.
Original synthetic final-stage illustration. The scale stays fixed. Cancellation at the central cluster is compatible with a low loss; a remote zero-score plateau has a much higher loss.
Reading a robust factor fit
The final refinement is commonly computed by iteratively reweighted least squares, starting from the robust fit. MASS warns that redescending losses can have multiple local minima. robustbase's fitting documentation likewise specifies an S start followed by a fixed-scale redescending M step and reports convergence separately. Re-estimating scale during that last step changes the procedure being described here.
A low weight is a statement about compatibility with the fitted model. It does not establish that a crisis observation is erroneous. In financial data, examine whether the fit is identifying a stable central relation or suppressing evidence of a changed regime; residual size alone also does not describe leverage.
SelfMadeQuant's Residual Diagnostics and Model Checking explains leverage, residual patterns and influence diagnostics that complement this examination. Its public page is a preview; the full lesson requires membership.
MASS, Robust Fitting of Linear Models, Arguments and Details: specific S-initialised MM construction, fixed scale, normal-model efficiency and local minima.
robustbase reference manual, lmrob and lmrob.fit, Description, Details and Value: MM-type implementation, fixed-scale final step and convergence reporting.
