Explicit solutions for the asymptotically optimal bandwidth in cross-validation

• Published in: Biometrika Vol. 111; no. 3; pp. 809 - 823
• Main Authors: Abadir, Karim M, Lubrano, Michel
• Format: Journal Article
• Language: English
• Published: Oxford Oxford University Press 01-09-2024; Oxford Publishing Limited (England); Oxford University Press (OUP)
• Subjects: Asymptotic methods; Bivariate analysis; Colleges & universities; Density; Economics and Finance; Humanities and Social Sciences; Multivariate analysis; Polynomials; Academic wage distribution; Nonparametric density estimation; Bandwidth choice; Explicit analytical solution; Cross-validation
• ISSN: 0006-3444; 1464-3510
• DOI: 10.1093/biomet/asae007

Abstract
We show that least-squares cross-validation methods share a common structure that has an explicit asymptotic solution, when the chosen kernel is asymptotically separable in bandwidth and data. For density estimation with a multivariate Student-t(ν) kernel, the cross-validation criterion becomes asymptotically equivalent to a polynomial of only three terms. Our bandwidth formulae are simple and noniterative, thus leading to very fast computations, their integrated squared-error dominates traditional cross-validation implementations, they alleviate the notorious sample variability of cross-validation and overcome its breakdown in the case of repeated observations. We illustrate our method with univariate and bivariate applications, of density estimation and nonparametric regressions, to a large dataset of Michigan State University academic wages and experience.