On optimal standard kernels. (English) Zbl 1159.62306

Summary: B. L. Granovsky and H. G. Müller [Int. Stat. Rev. 59, No. 3, 373–388 (1991; Zbl 0749.62024)] have shown that kernels of order \((v,k)\) with \(k-v\) even, i.e., functions which have a compact support and fulfil a certain moment condition, are optimal in some sense only if they are continuous and coincide on the support with a certain polynomial. In his diploma thesis, C. Pfeifer [Zur Theorie der optimalen Kernschätzer unter Momentenbedingungen. Dpt. Math., Univ. Marburg, Germany (1991)] proved that usually there are finitely many kernels having this property with support \([-1,a_i],\;i=1,\dots,m\). In this paper we show that in the “standard case” \(v=0\) the kernel with the largest support in that family is optimal.


62G07 Density estimation


Zbl 0749.62024