Cross-validation Revisited. Issue 2 (7th February 2016)

• Record Type: Journal Article
• Authors: Dutta, Santanu
• Is Part Of: Communications in statistics. Volume 45:Issue 2(2016)
• Page Start: 472
• Page End: 490
• Publication Date: 2016-02-07
• Subjects: Density estimation -- Least-squares cross-validation -- Pseudo-likelihood.; 62G07; Mathematical statistics -- Periodicals; Mathematical statistics -- Data processing -- Periodicals; Digital computer simulation -- Periodicals; 519.5
• DOI: 10.1080/03610918.2013.862275
• Languages: English
• ISSNs: 0361-0918

Abstract

Abstract : Data-based choice of the bandwidth is an important problem in kernel density estimation. The pseudo-likelihood and the least-squares cross-validation bandwidth selectors are well known, but widely criticized in the literature. For heavy-tailed distributions, the L 1 distance between the pseudo-likelihood-based estimator and the density does not seem to converge in probability to zero with increasing sample size. Even for normal-tailed densities, the rate of L 1 convergence is disappointingly slow. In this article, we report an interesting finding that with minor modifications both the cross-validation methods can be implemented effectively, even for heavy-tailed densities. For both these estimators, the L 1 distance (from the density) are shown to converge completely to zero irrespective of the tail of the density. The expected L 1 distance also goes to zero. These results hold even in the presence of a strongly mixing-type dependence. Monte Carlo simulations and analysis of the Old Faithful geyser data suggest that if implemented appropriately, contrary to the traditional belief, the cross-validation estimators compare well with the sophisticated plug-in and bootstrap-based estimators.