Bias correction and higher order kernel functions

Jianqing Fan Tien-Chung Hu

Statistics Theory and Methods mathscidoc:1912.43378

Statistics & probability letters, 13, (3), 235-243, 1992.2
Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(<i>h</i><sup>2</sup><sub><i>n</i></sub>). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function <i>K</i>, the functions W<sub>k</sub>(x)=<sup>k-1</sup><sub>1=0</sub>(<sup>k</sup><sub>l+1</sub>)x<sup>l</sup>K<sup>(l)</sup>(x)/l! and [1/<sup></sup><sub></sub><i>K</i><sup>(<i>k</i> 1)</sup>(<i>x</i>)/<i>x</i> d <i>x</i>]<i>K</i><sup>(<i>k</i> 1)</sup>(<i>x</i>)/<i>x</i> are kernels of order <i>k</i>, under some mild conditions.
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  title={Bias correction and higher order kernel functions},
  author={Jianqing Fan, and Tien-Chung Hu},
  booktitle={Statistics &amp; probability letters},
Jianqing Fan, and Tien-Chung Hu. Bias correction and higher order kernel functions. 1992. Vol. 13. In Statistics &amp; probability letters. pp.235-243.
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