Smooth change point estimation in regression models with random design

We consider the problem of estimating the location of a change point $$\theta _0$$ θ 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 $$\theta _0$$ θ 0 . The degree of smoothness $$q_0$$ q 0 has to be estimated as well. We investigate the consistency with increasing sample size $$n$$ n of the least squares estimates $$(\hat{\theta }_n,\hat{q}_n)$$ ( θ ^ n , q ^ n ) of $$(\theta _0, q_0)$$ ( θ 0 , q 0 ) . It turns out that the rates of convergence of $$\hat{\theta }_n$$ θ ^ n depend on $$q_0$$ q 0 : for $$q_0$$ q 0 greater than $$1/2$$ 1 / 2 we have a rate of $$\sqrt{n}$$ n and the asymptotic normality property; for $$q_0$$ q 0 less than $$1/2$$ 1 / 2 the rate is $$\displaystyle n^{1/(2q_0+1)}$$ n 1 / ( 2 q 0 + 1 ) and the change point estimator converges to a maximizer of a Gaussian process; for $$q_0$$ q 0 equal to $$1/2$$ 1 / 2 the rate is $$\sqrt{n \cdot \mathrm{ln}(n)}$$ n · ln ( n ) . Interestingly, in the last case the limiting distribution is also normal. Copyright The Institute of Statistical Mathematics, Tokyo 2015