The Annals of Statistics

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

Adam T. Martinsek

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Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348657

Digital Object Identifier
doi:10.1214/aos/1176348657

Mathematical Reviews number (MathSciNet)
MR1165593

Zentralblatt MATH identifier
0746.62041

JSTOR
links.jstor.org

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1176348657


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