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Optimal kernels. (English) Zbl 1159.62020

Summary: Kernel functions \(K(x)\) are widely used for smoothing purposes in statistics, for instance see B. W. Silverman [Density estimation for statistics and data analysis. Chapman and Hall (1986; Zbl 0617.62042)], or M. P. Wand and M. C. Jones [Kernel smoothing. Chapman and Hall (1995; Zbl 0854.62043)], and also M. G. Schimek [Glättungsverfahren in der Biometrie: ein historischer Abriss. Electron. J. GMS, Med. Inform. Biom. Epidemiol (2005)]. Usually, they have compact support and are of order \((\nu,k)\), which means that the \(j\)-th moment \(M_j\) is zero for \(j < k\) except \(j =\nu < k\) and is standardized appropriately for \(j = \nu.\) Here, \(M_j\) means the integral over the product of \(K(x)\) and \(x^j\). In the case of \(\nu = 0\), \(K\) is called a standard kernel. Kernels of order \((\nu,k)\) are called optimal if they change the sign exactly \(k-2\) times and minimize the asymptotical integrated mean square error. In the case of \(k-\nu\) being even, T. Gasser et al. [J. R. Stat. Soc., Ser. B 47, 238–252 (1985; Zbl 0574.62042)] 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 \((\nu,k)\). In some special cases those \(K\) could be proved to be optimal. Later on B. L. Granovsky and H.-G. Müller [Int. Stat. Rev. 59, No. 3, 373–388 (1991; Zbl 0749.62024)] 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 \(k-\nu < 4\). H.-G. Pfeifer [Zur Theorie der optimalen Kernschätzer unter Momentenbedingungen. Dept. Math., Univ. Marburg (1991)] in his diploma thesis constructed polynomials \(p_i(x)\) of degree \(k\) and reals \(\alpha_i\) in \([0,1]\) with \(\alpha_1 <\cdots < \alpha_m\) such that the restriction \(K_i\) of \(p_i\) to \([-1,\alpha_i]\) is of order \((\nu,k)\) and fulfills the boundary condition \(p_i(-1) = 0 = p_i(\alpha_i)\), \(i = 1,\dots,m\), where \(m\) is the integer part of \((k-\nu)/2\).
We prove a long-standing conjecture in its most general form, which in the standard case has been verified in one of our earlier papers [Stat. Decis. 19, No. 1, 1–8 (2001; Zbl 1159.62306)]. By means of the theory of Gegenbauer (ultraspherical) polynomials we show that in the general case of arbitrary \(k\), \(\nu\) with \(0\leq \nu\leq k - 2\) the kernel \(K_m\) is optimal and give its explicit form.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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