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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.

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