Estimating a gradual parameter change in an AR(1)-process

Abstract We discuss the estimation of a change-point $$t_0$$ t 0 at which parameter (non-stationary) AR(1)-process possibly changes in gradual way. Making use observations $$X_1,\ldots ,X_n$$ X 1 , … n , we shall study least squares estimator $$\widehat{t}_0$$ ^ for is obtained by minimizing sum residuals with respect to given parameters. As first result it can be shown that, under certain regularity and moment assumptions, $$\widehat{t}_0/n$$ / consistent $$\tau _0$$ ? where $$t_0 =\lfloor n\tau _0\rfloor $$ = ? ? $$0<\tau _0<1$$ < i.e., $$\widehat{t}_0/n \,{\mathop {\rightarrow }\limits ^{P}}\,\tau ? P $$(n\rightarrow \infty )$$ ( ? ) . Based on rates proof consistency result, first, but rough, convergence rate statement immediately given. Under somewhat stronger precise derived via asymptotic normality our estimator. Some results from small simulation are included give an idea finite sample behaviour proposed