Weighted Local Regression and Kernel Methods for Nonparametric Curve Fitting

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Abstract
Weighted local regression, a popular technique for smoothing scatterplots, is shown to be asymptotically equivalent to certain kernel smoothers. Since both methods are local weighted averages of the data, it is proved that in the fixed design regression model, given a weighted local regression procedure with any weight function, there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity. It is demonstrated by examples that in some instances the weights are nearly the same for both methods, even for small samples. The asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression. Further, a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order (number of vanishing moments) of the kernel employed in the kernel smoother and between the shapes of the kernel and the weight function.

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