Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation

Abstract

The performance of multivariate kernel density estimates depends crucially on the choice of bandwidth matrix, but progress towards developing good bandwidth matrix selectors has been relatively slow. In particular, previous studies of cross-validation (CV) methods have been restricted to biased and unbiased CV selection of diagonal bandwidth matrices. However, for certain types of target density the use of full (i.e. unconstrained) bandwidth matrices offers the potential for significantly improved density estimation. In this paper, we generalize earlier work from diagonal to full bandwidth matrices, and develop a smooth cross-validation (SCV) methodology for multivariate data. We consider optimization of the SCV technique with respect to a pilot bandwidth matrix. All the CV methods are studied using asymptotic analysis, simulation experiments and real data analysis. The results suggest that SCV for full bandwidth matrices is the most reliable of the CV methods. We also observe that experience from the univariate setting can sometimes be a misleading guide for understanding bandwidth selection in the multivariate case. Copyright 2005 Board of the Foundation of the Scandinavian Journal of Statistics..

A non-parametric test for industrial specialization

“…Second, since we incorporate downtown Denver as well as the surrounding areas such as Boulder and Greeley, and the Denver tech center, the study area is not single peaked. 16 Therefore, we estimate H using the smoothed cross validation (SCV) technique introduced by Hall et al (1992), which has been shown by Duong and Hazelton (2003) and Duong and Hazelton (2005b) to have a low Mean Integrated Square Error (MISE) for a range of target density shapes, an excellent convergence rate for small sample sizes, and an ability to accurately estimate the off-diagonal elements of the bandwidth matrix. The SCV bandwidth selection procedure is more formally discussed in Appendix 6.…”

Properties of Transmetric Density Estimation

“…It is straightforward to see that UCV for the common kernel density estimator, with a uniform kernel as of Duong and Hazelton (2005) is given by…”

“…In normal kernel density estimation, there are generally two methods that have received some attention, namely plug-in methods (Duong and Hazelton, 2003). and cross-validation methods (Duong and Hazelton, 2005). The plug-in methods require an analytic expression of the asymptotic mean square error.…”

“…In Duong and Hazelton (2005), the unbiased, the biased and the smoothed cross-validation methods are described and compared. The biased cross-validation methods requires The i'th component of x.…”

Social inequality and the biological standard of living: An anthropometric analysis of Swiss conscription data, 1875–1950

“…Härdle et al (2004)). In particular we employed the formulation of SCV method established by Duong and Hazelton (2005) and the algorithm proposed (an implemented) by Duong (2007). Besides the identification of features from a bivariate distribution it is of particular interest to quantify the directions in which these lay.…”