Abstract
In this paper we propose a new nonparametric regression technique. Our proposal has common ground with existing two-step procedures in that it starts with a parametric model. However, our approach differs from others in the choice of parametric start within the parametric family. Our proposal chooses a function that is the projection of the unknown regression function onto the parametric family in a certain metric, while the existing methods select the best approximation in the usual $L_{2}$ metric. We find that the difference leads to substantial improvement in the performance of regression estimators in comparison with direct one-step estimation, irrespective of the choice of a parametric model. This is in contrast with the existing two-step methods, which fail if the chosen parametric model is largely misspecified. We demonstrate this with sound theory and numerical experiment.
Highlights
We study a new approach to nonparametric regression
The conventional local linear estimator of m with a bandwidth b has the asymptotic bias b2cKm00(x)/2 with a constant cK depending on the kernel of the local linear smoother, while our new approach based on the decomposition (1.3) gives b2cKm000(x)/2, see Proposition 1
Hjort and Glad (1995), Glad (1998), Gozalo and Linton (2000), Rahman and Ullah (2002), Fan et al (2009) and Talamakrouni et al (2015, 2016). All these papers considered the approach that finds a pilot estimator of a parametric model assuming that the chosen parametric model is correct, and updates the parametric fit by a nonparametric adjustment
Summary
We study a new approach to nonparametric regression. Let m = E (Y |X = ·) denote the true regression function and we assume that m is twice continuously di↵erentiable with E m00(X)2 < 1. The conventional local linear estimator of m with a bandwidth b has the asymptotic bias b2cKm00(x)/2 with a constant cK depending on the kernel of the local linear smoother, while our new approach based on the decomposition (1.3) gives b2cKm000(x)/2, see Proposition 1. Hjort and Glad (1995), Glad (1998), Gozalo and Linton (2000), Rahman and Ullah (2002), Fan et al (2009) and Talamakrouni et al (2015, 2016) All these papers considered the approach that finds a pilot estimator of a parametric model assuming that the chosen parametric model is correct, and updates the parametric fit by a nonparametric adjustment.
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