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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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