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

Abstract

This paper considers M-estimation of a nonlinear regression model with multiple change-points occuring at 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 locations of 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.