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Bias correction and higher order kernel functions. (English) Zbl 0741.62048
Summary: Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of $$O(h_ n^ 2)$$. 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 $$K$$, the functions $W_ k(x)=\sum_{l=0}^{k-1}{k \choose l+1}x^ lK^{(l)}(x)/l!\quad\hbox{and}\quad[1/\int_{- \infty}^ \infty K^{(k-1)}(x)(x)^{-1} dx]K^{(k-1)}(x)/x$ are kernels of order $$k$$, under some mild conditions.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation
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