Abstract
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 \(\displaystyle n^{1/(2q_0+1)}\) and the change point estimator converges to a maximizer of a Gaussian process; for \(q_0\) equal to \(1/2\) the rate is \(\sqrt{n \cdot \mathrm{ln}(n)}\). Interestingly, in the last case the limiting distribution is also normal.
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The authors would like to thank the Associate Editor and Reviewers for their careful reading and comments. These comments and suggestions have been very helpful for revising and improving the manuscript.
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Döring, M., Jensen, U. Smooth change point estimation in regression models with random design. Ann Inst Stat Math 67, 595–619 (2015). https://doi.org/10.1007/s10463-014-0467-8
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DOI: https://doi.org/10.1007/s10463-014-0467-8