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June, 1992 Using Stopping Rules to Bound the Mean Integrated Squared Error in Density Estimation
Adam T. Martinsek
Ann. Statist. 20(2): 797-806 (June, 1992). DOI: 10.1214/aos/1176348657


Suppose $X_1,X_2,\ldots,X_n$ 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 sample size to solve this problem. However, the appropriate sample size depends on a functional of the density that typically is unknown. In this paper, 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.


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Adam T. Martinsek. "Using Stopping Rules to Bound the Mean Integrated Squared Error in Density Estimation." Ann. Statist. 20 (2) 797 - 806, June, 1992.


Published: June, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0746.62041
MathSciNet: MR1165593
Digital Object Identifier: 10.1214/aos/1176348657

Primary: 62G07
Secondary: 62G20, 62L12

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 2 • June, 1992
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