Function Estimation Using Data Adaptive Kernel Estimation - How Much Smoothing?

TL;DR

This paper introduces a data-adaptive kernel smoothing method that optimizes the kernel shape to minimize expected error, automatically adjusting to the unknown function for improved estimation accuracy.

Contribution

It proposes a novel scheme for automatically tuning kernel smoothers based on minimizing an estimated mean square error, incorporating penalty adjustments for data reuse.

Findings

The method effectively adapts to unknown functions.

Goodness of fit estimators are simple but sensitive to perturbations.

Fitting the mean square error model improves the stability of the optimal bandwidth.

Abstract

We determine the expected error by smoothing the data locally. Then we optimize the shape of the kernel smoother to minimize the error. Because the optimal estimator depends on the unknown function, our scheme automatically adjusts to the unknown function. By self-consistently adjusting the kernel smoother, the total estimator adapts to the data. Goodness of fit estimators select a kernel halfwidth by minimizing a function of the halfwidth which is based on the average square residual fit error: $ASR(h)$. A penalty term is included to adjust for using the same data to estimate the function and to evaluate the mean square error. Goodness of fit estimators are relatively simple to implement, but the minimum (of the goodness of fit functional) tends to be sensitive to small perturbations. To remedy this sensitivity problem, we fit the mean square error %goodness of fit functional to a two parameter model prior to determining the optimal halfwidth. Plug-in derivative estimators estimate the second derivative of the unknown function in an initial step, and then substitute this estimate into the asymptotic formula.