Optimal kernels

Summary 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 ( v , k ), which means that the j -th moment M j is zero for j < k except j = v < k and is standardized appropriately for j = v . Here, M j means the integral over the product of K ( x ) and x j . In the case of v = 0, K is called a standard kernel. Kernels of order ( v , 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 – v 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 ( v , 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 - v < 4. Pfeifer [19] in his diploma thesis constructed polynomials p i ( x ) of degree k and reals α i in [0,1] with α 1 < … < α m such that the restriction K i of p i to [-1, α i ] is of order ( v , k ) and fulfills the boundary condition p i (-1) = 0 = p i ( α i ), i = 1,…, m , where m is the integer part of ( k - v )/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 , v with 0 ≤ v ≤ k – 2 the kernel K m is optimal and give its explicit form.