Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices

Bowman AW (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71:353–360

Chacón JE (2009) Data-driven choice of the smoothing parametrization for kernel density estimators. Can J Stat 37:249–265

Chacón JE, Duong T, Wand MP (2009) Asymptotics for general multivariate kernel density derivative estimators (submitted)

Duong T (2009) ks: Kernel smoothing. R package version 1.6.5

Duong T, Hazelton ML (2003) Plug-in bandwidth matrices for bivariate kernel density estimation. J Nonparametr Stat 15:17–30

Duong T, Hazelton ML (2005a) Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation. J Multivar Anal 93:417–433

Duong T, Hazelton ML (2005b) Cross-validation bandwidth matrices for multivariate kernel density estimation. Scand J Stat 32:485–506

Hall P, Marron JS (1987) Estimation of integrated squared density derivatives. Stat Probab Lett 6:109–115

Hall P, Marron JS, Park BU (1992) Smoothed cross-validation. Probab Theory Relat Fields 92:1–20

Henderson HV, Searle SR (1979) Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. Can J Stat 7:65–81

Holmquist B (1985) The direct product permuting matrices. Linear Multilinear Algebra 17:117–141

Holmquist B (1996a) The d-variate vector Hermite polynomial of order k. Linear Algebra Appl 237–238:155–190

Holmquist B (1996b) Expectations of products of quadratic forms in normal variables. Stoch Anal Appl 14:149–164

Jones MC, Sheather SJ (1991) Using nonstochastic terms to advantage in kernel-based estimation of integrated squared density derivatives. Stat Probab Lett 11:511–514

Magnus JR, Neudecker H (1980) The elimination matrix: some lemmas and applications. SIAM J Algebra Discrete Methods 1:422–449

Magnus JR, Neudecker H (1999) Matrix differential calculus with applications in statistics and econometrics, revised edn. Wiley, Chichester

Meijer E (2005) Matrix algebra for higher order moments. Linear Algebra Appl 410:112–134

Park BU, Marron JS (1990) Comparison of data-driven bandwidth selectors. J Am Stat Assoc 85:66–72

Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27:832–837

Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Stat 9:65–78

Sain SR, Baggerly KA, Scott DW (1994) Cross-validation of multivariate densities. J Am Stat Assoc 89:807–817

Schott JR (2003) Kronecker product permutation matrices and their application to moment matrices of the normal distribution. J Multivar Anal 87:177–190

Schott JR (2005) Matrix analysis for statics, 2nd edn. Wiley, New York

Scott DW, Terrell GR (1987) Biased and unbiased cross-validation in density estimation. J Am Stat Assoc 82:1131–1146

Sheather SJ, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J R Stat Soc Ser B Stat Methodol 53:683–690

Silverman BW (1986) Density estimation for statics and data analysis. Chapman & Hall, London

Simonoff JS (1996) Smoothing methods in statics. Springer, Berlin

Wand MP (1992) Error analysis for general multivariate kernel estimators. J Nonparametr Stat 2:2–15

Wand MP, Jones MC (1993) Comparison of smoothing parameterizations in bivariate kernel density estimation. J Am Stat Assoc 88:520–528

Wand MP, Jones MC (1994) Multivariate plug-in bandwidth selection. Comput Stat 9:97–117

Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, London