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.
Change-points are intrinsic features of signals that appear in economics, medicine and physical science. The statistics literature contains a vast amount of works on issues related to the estimation of the change-point for a parametric regression, most of it specifically designed for the case of a single break. The more used estimators are the maximum likelihood estimators, the least squares estimators or a wider class, the M-estimators. Statistical inference for a parametrical model is influenced by the continuity or by discontinuity of the regression function at the change-points, but also by the determinist character or not of the explicative variable. We give a non-exhaustive list with the recent papers. The area of research is so active that it is nearly impossible to list all the recent papers written. For the least squares (LS) estimators we refer to Feder (1975aFeder ( , 1975b) ) for continuous two-lines models, Lai et al. (1979), Yao and Au (1989) for a step function, Liu et al. (1997), Bai and Perron (1998) for multiple structural changes in a linear model. For the maximum likelihood (ML) estimator, when the design is determinist, Bhattacharya (1994) discusses his limiting behaviour for a discontinuous linear model. Gill (2004), Gill and Baron (2004) consider a model where the canonical parameter of an exponential family gradually begins to drift from its initial value at an unknown change-point. For a random design we refer to Koul and Qian (2002) for two lines model, Ciuperca (2004) for a single jump in a nonlinear model, Ciuperca and Dapzol (2008) for multiple change-points in linear and nonlinear model. If the model variance depends of the mean, the quasi-likelihood estimator can be considered. Braun et al. (2000) consider that the mean is constant between two change-points. Chiou and Muller (2004) propose a semi-parametric estimator in a generalized linear model with determinist design. In the general case of M-estimators, Rukhin and Vajda (1997) consider the change-point estimation problem as a nonlinear regression problem, the model being continuous, with a single change-point and fixed design. Koul et al. (2003) study the M-estimators in two-phase linear regression with random design. The present paper makes several contributions to the existing literature. The considered design is random, the regression function is nonlineary within the framework of a multi-regime and not lastly, a general method of estimation. We study the properties of the M-estimator in a multi-phase discontinuous nonlinear random regression model with a general error distribution. The class of the M-estimators was introduced by Huber (1964) and its principal properties are exposed in Huber (1981). We generalize among others, the results for the two-phase random linear model of Koul et al. (2003) obtained by M-estimation, the results obtained by the ML estimation of Ciuperca and Dapzol (2008) for a multiphase random nonlinear model and of Bai and Perron (1998) obtained by LS estimation in a multiple nonrandom linear regression. An important point of the proofs for the linear case is the relation between the regression function and its derivatives with respect to regression parameters. Thus we have to modify the approach for the non linear regression. Also, in the case of a single change-point, each of two regimes has one fixed boundary. For multiple breaks, each middle regime has boundaries completely unknown. The paper is organized as follows. We give necessary notations and definitions in Section 2. In Section 3 we establish the estimators consistency and the convergence rate. Weak convergence results are also obtained: the asymptotic distribution of the regression parameters M-estimator is Gaussian. We also prove that n( θ2nθ 0 2 ) converges weakly to the smallest minimizer vector of the independent compound Poisson processes, where θ2n is the change-point estimator. Auxiliary results are given in Appendix.
Consider the step-function with K (K ≥ 1) fixed change-points, for x ∈ IR:
where θ 1 = (α 0 , α 1 , …., α K ) are the nonlinear regression parameters and θ 2 = (τ 1 , …, τ K ), τ 1 < τ 2 < … < τ K are the change-points. For all k = 0, 1, …, K, we have the parameter α k belongs to some compact Γ ⊆ IR d . We consider that the vector θ 2 ∈ IR K and we set θ = (θ 1 , θ 2 ) ∈ Ω = Γ K+1 × IR K . Consider the random design model:
where (ε i , X i ) is a sequence of continuous independent random variables with the same joint distribution as (ε, X). The parameter θ 1 and the change-points (or break points) are unknown.
The purpose is to estimate θ = (θ 1 , θ 2 ) when n observations of (Y, X) are available. We denote the true value of a parameter with a 0 superscript. In particular, θ 0 1 = (α 0 0 , α 0 1 , …, α 0 K ) and θ 0 2 = (τ 0 1 , …, τ 0 K ) are used to denote, respectively, the true values of the regression parameters and the true change-points. Let be also θ 0 = (θ 0 1 , θ 0 2 ). We suppose that θ
