Nonparametric kernel regression estimation near endpoints

Abstract When kernel regression is used to produce a smooth estimate of a curve over a finite interval, boundary problems detract from the global performance of the estimator. A new kernel is derived to reduce this boundary problem. A generalized jackknife combination of two unsatisfactory kernels produces the desired result. One motivation for adopting a jackknife combination is that they are simple to construct and evaluate. Furthermore, as in other settings, the bias reduction property need not cause an inordinate increase in variability. The convergence rate with the new boundary kernel is the same as for the non-boundary. To illustrate the general approach, a new second-order boundary kernel, which is continuously linked to the Epanechnikov (1969, Theory Probab. Appl. 14, 153–158) kernel, is produced. The asymptotic mean square efficiencies relative to smooth optimal kernels due to Gasser and Muller (1984, Scand. J. Statist. 11, 171–185), Muller (1991, Biometrika 78, 521–530) and Muller and Wang (1994, Biometrics 50, 61–76) indicate that the new kernel is also competitive in this sense.

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