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Cross-validation bandwidth matrices for multivariate kernel density estimation. (English) Zbl 1089.62035
A \(d\)-variate kernel estimator for a density \(f\) of an i.i.d. sample \(X_i\), \(i=1,\dots,n\), is considered being of the form \[ \hat f(x;H)=n^{-1}(\det H)^{-1/2}\sum_{i=1}^n K(H^{-1/2}(x-X_i)), \] where \(H\) is the bandwidth matrix, and \(K\) is a kernel function. The authors consider three cross-validation (CV) techniques of \(H\) selection: unbiased CV targeting exact mean integrated squared error (MISE) minimization, biased CV based on the estimation of the asymptotic MISE, and smooth CV which uses data smoothed by a pilot kernel smoother. The asymptotic behaviour of the selectors is investigated. The obtained estimates are compared via simulations to plug-in selectors and selectors with diagonal bandwidth matrices. An application to bivariate demographic data is considered. The authors’ conclusion is that “CV for full bandwidth matrices is the most reliable method among these CV selectors that we studied. For bivariate data …it is reasonably comparable to the best plug-in methods currently available”.

MSC:
62G07 Density estimation
62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
62G09 Nonparametric statistical resampling methods
Software:
KernSmooth; pyuvdata; R
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References:
[1] Bowman A., Biometrika 71 pp 353– (1984)
[2] DOI: 10.1080/10485250306039 · Zbl 1019.62032 · doi:10.1080/10485250306039
[3] Duong T., J. Multivariate Anal. (2004)
[4] DOI: 10.1007/BF01205233 · Zbl 0742.62042 · doi:10.1007/BF01205233
[5] Hazelton M., Scand. J. Statist. 23 pp 221– (1996)
[6] Ihaka R., J. Comput. Graph. Statist. 5 pp 299– (1996)
[7] DOI: 10.1016/0167-7152(92)90107-G · doi:10.1016/0167-7152(92)90107-G
[8] Jones M., Scand. J. Statist. 19 pp 337– (1992)
[9] Jones M., Ann. Statist. 19 pp 1919– (1991)
[10] Jones M., Comput. Statist. 91 pp 337– (1996)
[11] Magnus J., Matrix differential calculus with applications in statistics and econometrics (1988) · Zbl 0651.15001
[12] Marron J., Comput. Statist. 8 pp 17– (1993) · doi:10.1063/1.45028
[13] Rudemo M., Scand. J. Statist. 9 pp 65– (1982)
[14] DOI: 10.1016/S0167-9473(01)00053-6 · Zbl 1132.62329 · doi:10.1016/S0167-9473(01)00053-6
[15] Sain S. R., J. Amer. Statist. Assoc. 89 pp 807– (1994)
[16] Scott D., Multivariate density estimation; theory, practice and visualization (1992) · Zbl 0850.62006 · doi:10.1002/9780470316849
[17] Scott D., J. Amer. Statist. Assoc. 82 pp 1131– (1987)
[18] Simonoff J., Smoothing methods in statistics (1996) · Zbl 0859.62035 · doi:10.1007/978-1-4612-4026-6
[19] Terrell G. R., J. Amer. Statist. Assoc. 85 pp 470– (1990)
[20] Wand M., Kernel smoothing (1995) · doi:10.1007/978-1-4899-4493-1
[21] Wand M. P., J. Amer. Statist. Assoc. 88 pp 520– (1993)
[22] Wand M. P., Comput. Statist. 9 pp 97– (1994)
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