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Using stopping rules to bound the mean integrated squared error in density estimation. (English) Zbl 0746.62041
Summary: Suppose \(X_ 1,X_ 2,\dots,X_ n\) are i.i.d. with unknown density \(f\). There is a well-known expression for the asymptotic mean integrated squared error (MISE) in estimating \(f\) by a kernel estimate \(\hat f_ n\), under certain conditions on \(f\), the kernel and the bandwidth. Suppose that one would like to choose a sample size so that the MISE is smaller than some preassigned positive number \(w\). Based on the asymptotic expression for the MISE, one can identify an appropriate sampel size to solve this problem. However, the appropriate sample size depends on a functional of the density that typically is unknown.
A stopping rule is proposed for the purpose of bounding the MISE, and this rule is shown to be asymptotically efficient in a certain sense as \(w\) approaches zero. These results are obtained for data-driven bandwidths that are asymptotically optimal as \(n\) goes to infinity.

62G07 Density estimation
62L12 Sequential estimation
62G20 Asymptotic properties of nonparametric inference
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