The M-estimator in a multi-phase random nonlinear model

This paper considers M-estimation of a nonlinear regression model with multiple change-points occuring at the unknown times. The multi-phase random design regression model, discontinuous in each change-point, have an arbitrary error $\epsilon$. In the case when the number of jumps is known, the M-estimator of the locations of the breaks and of regression parameters are studied. These estimators are consistent and the distribution of the regression parameter estimators is Gaussian. The estimator of each change-point converges, with the rate $n^{-1}$, to the smallest minimizer of the independent compound Poisson processes. The results are valid for a large class of error distributions.