Adaptive Order Polynomial Fitting: Bandwidth Robustification and Bias Reduction

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Abstract
This article deals with estimation of the regression function and its derivatives using local polynomial fitting. An important question is: How to determine the order of the polynomial to be fitted in a particular fixed neighborhood? This depends on the local curvature of the unknown curve. A higher order fit leads to a possible bias reduction, but results in an increase of variability. A precise evaluation of this increase is presented, and from this it is also clear that it is preferable to choose the order of fit adaptively. In this article we provide, for a given bandwidth, such a data-driven variable order selection procedure. The basic idea is to obtain a good estimate of the mean squared error at each location point and to use this estimate as a criterion for the order selection. The performance of the proposed selection procedure is illustrated via simulated examples. It turns out that the adaptive order fit is more robust against bandwidth variation; even if the bandwidth varies by a factor of 3, the resulting estimates are qualitatively indistinguishable. Hence the issue of choosing the bandwidth becomes less important and a crude bandwidth selector might suffice. We propose such a simple rule for selecting the bandwidth, and demonstrate its performance for the adaptive order fit via some simulated examples.

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