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

我们讨论了一个变化点\(t_0\)的估计，在这个点上（非平稳的）AR(1) 过程的参数可能以逐渐的方式变化。利用所述观测\（X_1，\ ldots，X_n \） ，我们将研究最小二乘估计\（\ widehat {吨} _0 \）为\（T_0 \） ，这是通过最小化的平方和得到的给定参数的残差。作为第一个结果，可以证明，在某些规律性和矩假设下，\(\widehat{t}_0/n\)是\(\tau _0\)的一致估计量，其中\(t_0 =\lfloor n \tau _0\rfloor \)，与\(0<\tau _0<1\)，即，\(\widehat{t}_0/n \,{\mathop {\rightarrow }\limits ^{P}}\,\tau _0\) \((n\rightarrow \infty )\)。基于在一致性结果证明中获得的速率，可以立即给出第一个但粗略的收敛速率声明。在稍微强一些的假设下，可以通过我们的估计量的渐近正态性推导出精确的速率。包括来自小型模拟研究的一些结果，以了解所提议估计器的有限样本行为。