The S-estimator in the change-point random model with long memory

This paper considers two-phase random design linear regression models. Errors and regressors are stationary long-range-dependent Gaussian processes. The regression parameters, the scale parameter and the change-point are estimated using a method introduced by Rousseeuw and Yohai [Robust regression by means of S-estimators, in Robust and Nonlinear Time Series Analysis, J. Franke, W. Hrdle, and R.D. Martin, eds., Lecture Notes in Statistics, Vol. 26, Springer, New York, 1984, pp. 256–272], which is called the S-estimator and has the property be more robust than the classical estimators in the sense that the outliers do not bias the estimation results. Some asymptotic results, including the strong consistency and the convergence rate of the S-estimator are proved. Simulations and an application to the Nile River data are also presented. It is shown via Monte Carlo simulations that the S-estimator is better than two other estimators that are proposed in the literature.