MM-Estimation and the Breakdown Point

In Lesson 1, the rental agency had one mistyped rent, and a robust fit shrugged it off. Toggle the buttons below from OLS to Huber and watch the least-squares line snap back off the single bad point. That is the win we are building on.

But it has a blind spot. Lesson 1's outlier sat in the middle of the size range, a plain vertical mistake. What happens when the bad rows sit far out along the x-axis, and there are many of them? In this lesson a batch of fraudulent listings will fool not just OLS but Huber and Tukey too, and we will build the estimator that still holds.

By the end of this lesson you will be able to:

• Explain how a bad high-leverage row keeps a small residual and so hides from a residual-based robust fit (masking)
• Define an estimator's breakdown point and say why 50% is the hard ceiling
• Explain how S-estimation reaches that ceiling, and how MM-estimation keeps the robustness while regaining precision
• Fit an MM regression in R with lmrob(), read its weights, and know when to reach for it over rlm()

Prerequisites: Lesson 1, Robust Regression with M-Estimators. You should know what an M-estimator, the Huber and Tukey weights, and rlm() are, and be comfortable telling leverage apart from influence.

A bad row that hides in plain sight

Lesson 1's typo was a vertical outlier: an ordinary floor area with a wrong rent, sitting well above or below the line in the thick of the data. Its large residual made it easy for Huber to spot.

This lesson's danger is different. A bad high-leverage point is unusual in two ways at once: its floor area is far from every other listing (high leverage, so it sits at the end of a long see-saw), and its rent is off the true trend. Drag the far-right point in the plot below down and away from the line. Notice what the solid "with the point" line does: instead of ignoring the stray point, it swings to chase it. And once the line has chased it, look at the point's residual, the vertical gap from the point to the line. It is small, because the line came to meet it.

That small residual is the whole problem. A robust method that judges a row by its residual will look at this chased-after point and see nothing wrong at all.