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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.

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