## The Annals of Statistics

### Consistent Cross-Validated Density Estimation

#### Abstract

Application of nonparametric density estimators generally requires the specification of a "smoothing parameter." The kernel estimator, for example, is not fully defined until a window width, or scaling, for the kernels has been chosen. Many "data-driven" techniques have been suggested for the practical choice of smoothing parameter. Of these, the most widely studied is the method of cross-validation. Our own simulations, as well as those of many other investigators, indicate that cross-validated smoothing can be an extremely effective practical solution. However, many of the most basic properties of cross-validated estimators are unknown. Indeed, recent results show that cross-validated estimators can fail even to be consistent for seemingly well-behaved problems. In this paper we will review the application of cross-validation to the smoothing problem, and establish $L_1$ consistency for certain cross-validated kernels and histograms.

#### Article information

Source
Ann. Statist., Volume 11, Number 1 (1983), 25-38.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346053

Digital Object Identifier
doi:10.1214/aos/1176346053

Mathematical Reviews number (MathSciNet)
MR684860

Zentralblatt MATH identifier
0509.62033

JSTOR