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Computationally efficient classes of higher‐order kernel functions
Author(s) -
Abdous Belkacem
Publication year - 1995
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315548
Subject(s) - kernel (algebra) , mathematics , fourier transform , kernel density estimation , polynomial , order (exchange) , selection (genetic algorithm) , variable kernel density estimation , fourier series , algorithm , kernel method , combinatorics , computer science , mathematical analysis , statistics , artificial intelligence , finance , estimator , support vector machine , economics
Classes of higher‐order kernels for estimation of a probability density are constructed by iterating the twicing procedure. Given a kernel K of order l , 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 —$(.)} m+1 , where Ǩ 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.