Asymptotic properties of semiparametric [formula omitted]-estimators with multiple change points

Abstract

Statistical models with multiple change points are used in many fields; however, the theoretical properties of semiparametric M-estimators of such models have received relatively little attention. The main purpose of the present work is to investigate the asymptotic properties of semiparametric M-estimators with non-smooth criterion functions of the parameters of a multiple change-point model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the semiparametric M-estimators of the change points is established and the rate of convergence is determined. The asymptotic normality of the semiparametric M-estimators of the parameters of the within-segment distributions is established under quite general conditions. These results, together with a generic paradigm for studying semiparametric M-estimators with multiple change points, provide a valuable extension to previous related research on (semi)parametric maximum-likelihood estimators. For illustration, the classification with missing data in the model is investigated in detail and a short simulation result is provided.