Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation. (English) Zbl 1066.62059

Summary: Progress in selection of smoothing parameters for kernel density estimation has been much slower in the multivariate than univariate setting. Within the context of multivariate density estimation attention has focused on diagonal bandwidth matrices. However, there is evidence to suggest that the use of full (or unconstrained) bandwidth matrices can be beneficial. This paper presents some results in the asymptotic analysis of data-driven selectors of full bandwidth matrices. In particular, we give relative rates of convergence for plug-in selectors and a biased cross-validation selector.


62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation


pyuvdata; KernSmooth
Full Text: DOI


[1] Duong, T.; Hazelton, M.L., Plug-in bandwidth matrices for bivariate kernel density estimation, J. nonparametric statist., 1, 17-30, (2003) · Zbl 1019.62032
[2] T. Duong, M.L. Hazelton, Cross-validation bandwidth matrices for multivariate kernel density estimation (2004) submitted for publication. · Zbl 1089.62035
[3] Jones, M.C.; Kappenman, R.F., On a class of kernel density estimate bandwidth selectors, Scand. J. statist. theory appl., 19, 337-349, (1992) · Zbl 0767.62035
[4] Jones, M.C.; Marron, J.; Sheather, S., A brief survey of bandwidth selection for density estimation, J. amer. statist. assoc., 91, 401-407, (1996) · Zbl 0873.62040
[5] Jones, M.C.; Sheather, S.J., Using nonstochastic terms to advantage in kernel-based estimation of integrated squared density derivatives, Statist. probab. lett., 11, 511-514, (1991) · Zbl 0724.62040
[6] Magnus, J.R.; Neudecker, H., Matrix differential calculus with applications in statistics and econometrics, (1988), Wiley Chichester · Zbl 0651.15001
[7] Sain, S.R.; Baggerly, K.A.; Scott, D.W., Cross-validation of multivariate densities, J. amer. statist. assoc., 89, 807-817, (1994) · Zbl 0805.62059
[8] Scott, D.W., Multivariate density estimation: theory, practice and visualization, (1992), Wiley New York · Zbl 0850.62006
[9] Scott, D.; Terrell, G., Biased and unbiased cross-validation in density estimation, J. amer. statist. assoc., 82, 1131-1146, (1987) · Zbl 0648.62037
[10] Wand, M.P., Error analysis for general multivariate kernel estimators, J. nonparametric statist., 2, 1-15, (1992) · Zbl 1360.62141
[11] Wand, M.P.; Jones, M.C., Comparison of smoothing parameterizations in bivariate kernel density estimation, J. amer. statist. assoc., 88, 520-528, (1993) · Zbl 0775.62105
[12] Wand, M.P.; Jones, M.C., Multivariate plug-in bandwidth selection, Comput. statist., 9, 97-116, (1994) · Zbl 0937.62055
[13] Wand, M.P.; Jones, M.C., Kernel smoothing, (1995), Chapman & Hall London · Zbl 0854.62043
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