Classic Newton-Raphson estimation converges to $x$ using the relationship: $$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$ In their paper on robust estimators of location and scale for small sample sizes, Rousseeuw & Verboven (2002, §4.1) state without source: "It is well-known in the robustness folklore that it is safer to replace the denominator by the constant $\int \psi'(u)\;d\Phi(u)$."
I have not been able to find a source for this. Moreover, in an example I've tried, I found faster convergence using the true first derivative instead of the calculated integral. I'm interested in the source for, and discussions about, this statement. Thank you.
Update: 2020-06-08
I've used these estimates often enough that I created an R package, revss: Robust Estimation in Very Small Samples for them.
The issue is that redescending psi-functions have a negative derivative in certain regions, and so it could happen that you encounter a near-zero denominator in the iterations. For strictly monotone psi there should indeed be no problem.