Optimal Inference Methods in Linear Models with Change-points
In this dissertation, we consider an estimation problem of the regression coefficients in both multiple regression model and multivariate multiple regression model with several unknown change-points. More precisely, we consider the cases where the target parameters (vectors or matrices) are suspected to satisfy some restrictions. In particular, we generalize in four ways, some recent statistical methods in the literatures. First, we relax some assumptions about the dependance structure of the noise and the regressors given in recent literature. Namely, in this dissertation, the dependance structure is as weak as that of an L2-mixingale arrays of size -1/2. Second, under such weak assumptions, we derive the joint asymptotic normality between the RE and UE. Third, in the context of change-points models, the estimation problem of a vector of the regression coefficients is extended to that of a matrix of the regression coefficients. Fourth, we propose a class of estimators which includes as a special cases shrinkage estimators~(SEs) as well as the unrestricted estimator~(UE) and the restricted estimator~(RE). We also derive a more general condition for the SEs to dominate the UE in mean square error. To this end, we generalize some identities for the evaluation of the bias and risk functions of shrinkage-type estimators (vectors or matrices). As illustrative example, our method is applied to the "gross domestic product'' data set of 10 countries including USA, Canada, UK, France and Germany. The simulation results corroborate our theoretical findings.