## The Annals of Statistics

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

#### 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

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