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Smooth change point estimation in regression models with random design. (English) Zbl 1440.62128
Summary: We consider the problem of estimating the location of a change point $$\theta_0$$ in a regression model. Most change point models studied so far were based on regression functions with a jump. However, we focus on regression functions, which are continuous at $$\theta_0$$. The degree of smoothness $$q_0$$ has to be estimated as well. We investigate the consistency with increasing sample size $$n$$ of the least squares estimates $$(\hat{\theta}_n,\hat{q}_n)$$ of $$(\theta_0, q_0)$$. It turns out that the rates of convergence of $$\hat{\theta}_n$$ depend on $$q_0$$: for $$q_0$$ greater than $$1/2$$ we have a rate of $$\sqrt{n}$$ and the asymptotic normality property; for $$q_0$$ less than $$1/2$$ the rate is $$n^{1/(2q_0+1)}$$ and the change point estimator converges to a maximizer of a Gaussian process; for $$q_0$$ equal to $$1/2$$ the rate is $$\sqrt{n \cdot \ln (n)}$$. Interestingly, in the last case the limiting distribution is also normal.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference
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