Change-points in nonparametric regression analysis.

*(English)*Zbl 0783.62032Starting 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).

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).

Reviewer: V.R.Eastwood (Wolfville)

##### MSC:

62G07 | Density estimation |

60F05 | Central limit and other weak theorems |

62G20 | Asymptotic properties of nonparametric inference |