In this paper we consider a regression model that allows for time series covariates as well as heteroscedasticity with a regression function that is modelled nonparametrically. We assume that the regression function changes at some unknown time $$\lfloor ns_0\rfloor $$ ⌊ n s 0 ⌋ , $$s_0\in (0,1)$$ s 0 ∈ ( 0 , 1 ) , and our aim is to estimate the (rescaled) change point $$s_0$$ s 0 . The considered estimator is based on a Kolmogorov-Smirnov functional of the marked empirical process of residuals. We show consistency of the estimator and prove a rate of convergence of $$O_P(n^{-1})$$ O P ( n - 1 ) which in this case is clearly optimal as there are only n points in the sequence. Additionally we investigate the case of lagged dependent covariates, that is, autoregression models with a change in the nonparametric (auto-) regression function and give a consistency result. The method of proof also allows for different kinds of functionals such that Cramér-von Mises type estimators can be considered similarly. The approach extends existing literature by allowing nonparametric models, time series data as well as heteroscedasticity. Finite sample simulations indicate the good performance of our estimator in regression as well as autoregression models and a real data example shows its applicability in practise.

中文翻译：

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估计非参数时间序列回归模型中的变化点

在本文中，我们考虑了一个回归模型，该模型允许时间序列协变量以及具有非参数建模的回归函数的异方差性。我们假设回归函数在某个未知时间发生变化 $$\lfloor ns_0\rfloor $$ ⌊ ns 0 ⌋ , $$s_0\in (0,1)$$s 0 ∈ ( 0 , 1 ) ，我们的目标是估计（重新调整的）变化点 $$s_0$$s 0 。所考虑的估计量基于残差的标记经验过程的 Kolmogorov-Smirnov 函数。我们展示了估计器的一致性，并证明了 $$O_P(n^{-1})$$OP ( n - 1 ) 的收敛速度，在这种情况下，这显然是最优的，因为序列中只有 n 个点。此外，我们调查滞后相关协变量的情况，即，具有非参数（自动）回归函数变化的自回归模型并给出一致性结果。证明方法还允许不同种类的泛函，因此可以类似地考虑 Cramér-von Mises 类型估计量。该方法通过允许非参数模型、时间序列数据以及异方差性来扩展现有文献。有限样本模拟表明我们的估计器在回归和自回归模型中的良好性能，一个真实的数据示例显示了它在实践中的适用性。