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Computationally efficient classes of higher-order kernel functions. (English) Zbl 0819.62031
Summary: Classes of higher-order kernels for estimation of a probability density are constructed by iterating the twicing procedure. Given a kernel \(K\) of order \(I\), we build a family of kernels \(K_ m\) of orders \(l(m+1)\) with the attractive property that their Fourier transforms are simply \(1 - \{1 - \widehat{K}(\cdot)\}^{m + 1}\), where \(\widehat{K}\) is the Fourier transform of \(K\). These families of higher-order kernels are well suited when the fast Fourier transform is used to speed up the calculation of the kernel estimate or the least-squares cross-validation procedure for selection of the window-width. We also compare the theoretical performance of the optimal polynomial based kernels with that of the iterative twicing kernels constructed from some popular second-order kernels.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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