The Annals of Statistics

Using Stopping Rules to Bound the Mean Integrated Squared Error in Density Estimation

Adam T. Martinsek

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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.

Article information

Ann. Statist., Volume 20, Number 2 (1992), 797-806.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62L12: Sequential estimation 62G20: Asymptotic properties

Density estimation stopping rule sequential estimation asymptotic efficiency mean integrated squared error


Martinsek, Adam T. Using Stopping Rules to Bound the Mean Integrated Squared Error in Density Estimation. Ann. Statist. 20 (1992), no. 2, 797--806. doi:10.1214/aos/1176348657.

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