## 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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