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Volker Mammitzsch
Optimal kernels
Keywords: smoothing kernels, polynomial kernels, asymptotic integrated mean square error
Kernel functions K(x) are widely used for smoothing purposes in statistics, for instance see Silverman [22] or Wand and Jones [24] and also Schimek [21]. Usually, they have compact support and are of order (ν,k), which means that the j-th moment Mj is zero for j < k except j = ν < k and is standardized appropriately for j = ν. Here, Mj means the integral over the product of K(x) and xj. In the case of ν = 0, K is called a standard kernel. Kernels of order (ν,k) are called optimal if they change the sign exactly k-2 times and minimize the asymptotical integrated mean square error. In the case of k-ν being even, Gasser et al. [3] have constructed polynomials K(x) of degree k with K(-1) = 0 = K(+1) which restricted to [-1,1] have exactly k-2 sign changes and are of order (ν,k). In some special cases those K could be proved to be optimal. Later on Granovsky and Müller [5] showed that optimal kernels are continuous functions with K(-1) = 0 = K(+1) and are polynomials on their support. Unfortunately, the converse is true only in the case k-ν < 4. Pfeifer [19] in his diploma thesis constructed polynomials pi(x) of degree k and reals αi in [0,1] with α1 < … < αm such that the restriction Ki of pi to [-1,αi] is of order (ν,k) and fulfills the boundary condition pi(-1) = 0 = pi(αi), i = 1,…,m, where m is the integer part of (k-ν)/2; see also Granovsky et al. [6]. In the present paper we prove a long-standing conjecture in its most general form, which in the standard case has been verified by Mammitzsch [14]. By means of the theory of Gegenbauer (ultraspherical) polynomials we show that in the general case of arbitrary k,ν with 0 ≤ ν ≤ k-2 the kernel Km is optimal and give its explicit form.
Statistics & Decisions, Oldenbourg Wissenschaftsverlag
Print ISSN: 0721-2631
Volume: 25, 02/2007
Pages: 153 - 172