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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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