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Optimal rate for estimating local bandwidth in kernel estimators of regression functions. (English) Zbl 0861.62031
Summary: This paper studies the best possible rates for estimating locally optimal bandwidth in kernel smoothing. Let \(b_0\) and \(\widehat{b}_0\) be the best data-adaptive bandwidths in the sense of mean squared error (MSE) and squared error (SE), respectively. It is shown that the best relative rate for estimating \(b_0\) is \(O_p(n^{-2/9})\) while that for estimating \(\widehat{b}_0\) is \(O_p(n^{2/45})\), which is much worse than that for global bandwidth selection. The results can be extended to other linear smoothing methods, to density estimations, and to derivative estimations. It is also shown that the bandwidth chosen according to an unbiased risk criterion approaches \(b_0\) at best at a relative rate \(n^{-1/15}\).

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference