Convergence Rate of Nonparametric Estimates of Maximum-Likelihood Type

The authors obtain the rate of convergence of $M$-estimates of nonparametric regression in the $L_2$ metric. It is shown that, for classes of smooth, monotonic, and convex functions, this rate cannot be improved (to within a constant). It is established that in a number of cases, particularly for the class of mono-tonic functions, nonlinear $M$-estimates are better than any linear estimates in terms of the order of the rate of convergence.