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Change-points in nonparametric regression analysis. (English) Zbl 0783.62032
Starting with a fixed design regression model $y_{i,n}= g(t_{i,n})+ \varepsilon_{i,n},\quad g\in{\mathcal L}^ \ell,\quad t_{i,n}\in [0,1],\qquad 1\leq i\leq n,$ with i.i.d. nonsystematic errors with common finite variance and equidistant design points, a possible changepoint of $$g^{(\nu)}$$, $$\nu\geq 0$$, is studied at unknown time $$\tau\in (0,1)$$. Investigated are the weak convergence of estimators $$\widehat\tau$$ of the time of change and rates of global $$L^ p$$-convergence of kernel estimators (adjusted to the estimated changepoint).
The main idea in finding estimators $$\widehat\tau$$ is to analyse the maximal occurring difference $$\widehat\Delta^{(\nu)}(t)$$ of the right- and left-sided regression estimates $$\widehat g^{(\nu)}_ \pm(t)$$. These regression estimates are based on one-sided kernel functions. Various kernel functions with asymmetric supports are considered. Weak convergence of the properly standardized estimator $\widehat \tau=\inf\left\{\rho\in Q:\;\widehat\Delta^{(\nu)}(\rho) = \sup_{x\in Q} \widehat \Delta^{(\nu)}(x)\right\}$ under certain regularity conditions to a normal random variable is proven. The rate of convergence here exceeds $$n^{-1/2}$$ in most cases. Asymptotic $$(1-\alpha) 100\%$$ confidence intervals are also derived for $$\widehat\tau$$ itself as well as for the size of the jump. Next the problem of global $$L^ p$$- consistency of kernel estimators under the presence of a changepoint at unknown time $$\tau$$ is discussed. Finally, the methods of the paper are illustrated on the annual volume of the Nile river (1871-1970).

##### MSC:
 62G07 Density estimation 60F05 Central limit and other weak theorems 62G20 Asymptotic properties of nonparametric inference
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