Kernel based nonparametric regression for discontinuous functions

Research Report SRR95-012

Kernel based nonparametric regression for discontinuous functions

Mark Bunt, Inge Koch, Alun Pope

Abstract: We consider the use of a segmentation strategy for estimating discontinuous functions from noisy data. The effect on mean integrated square error (MISE) pf kernel smoothing near a discontinuity is examined for the Nadaraya-Watson (NW) and the local linear (LL) estimator. We show that, provided a good discontinuity detector/locator is used and the data is segmented and estimated on each segment, the MISE of the LL estimator has an asymptotic rate of convergence as good as for smooth functions, and the LL estimator performs much better than the NW estimator. Furthermore, we show that the effect on MISE of an undetected (real) discontinuity in the data is of the same order as the kernel bandwidth, and thus dominates MISE, while adding a spurious discontinuity in the segmentation and estimation process leads to an (asymptotically) negligible contribution to MISE. Extensions of our results to the class of estimators which fit locally polynomials of arbitrary degree are given. These results show that discontinuous nonparametric regression performs as well asymptotically as ordinary nonparametric regression for smooth functions, provided an appropriate segmentation strategy is used.

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