Abstract
In this chapter, we get our first taste of real analysis, starting with some results on the approximations of functions in L 1. The literature on this is vast, and a lot of it is ancient. The problem is that f cannot be approximated in L 1 by μ n , the empirical measure, as the total variation distance between any density f and any atomic measure (like μ n ) is 1. Thus, the approximation itself must have a density. The kernel estimate provides this: it smooths the empirical measure μ n . This section has no combinatorial contributions, but develops just enough of the function approximation material to understand the remaining chapters.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
§9.12. References
H. Akaike, “An approximation to the density function,” Annals of the Institute of Statistical Mathematics, vol. 6, pp. 127–132, 1954.
J. Beirlant and D. M. Mason, “On the asymptotic normality of L p-norms of empirical functionals,” Mathematical Methods of Statistics, vol. 4, pp. 1–19, 1995.
A. Berlinet and L. Devroye, “A comparison of kernel density estimates,” Publications de l’Institut de Statistique de l’Université de Paris, vol. 38, pp. 3–59, 1994.
L. Breiman, W. Meisel, and E. Purcell, “Variable kernel estimates of multivariate densities,” Technometrics, vol. 19, pp. 135–144, 1977.
M. Broniatowski, P. Deheuvels, and L. Devroye, “On the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate,” Annals of Statistics, vol. 17, pp. 1070–1086, 1989.
R. Cao, A. Cuevas, and W. González-Manteiga, “A comparative study of several smoothing methods in density estimation,” Computational Statistics and Data Analysis, vol. 17, pp. 153–176, 1994.
R. Cao and L. Devroye, “The consistency of a smoothed minimum distance estimate,” Scandinavian Journal of Statistics, vol. 23, pp. 405–418, 1996.
D. B. H. Cline, “Admissible kernel estimators of a multivariate density,” Annals of Statistics, vol. 16, pp. 1421–1427, 1988.
D. B. H. Cline, “Optimal kernel estimation of densities,” Annals of the Institute of Statistical Mathematics, vol. 42, pp. 287–303, 1990.
M. Csörgő and L. Horváth, “Central limit theorems for L p-norms of density estimators,” Probability Theory and Related Fields, vol. 80, pp. 269–291, 1988.
P. Deheuvels, “Estimation nonparamétrique de la densité par histogrammes generalisés,” Publications de l’Institut de Statistique de l’Université de Paris, vol. 22, pp. 1–23, 1977.
L. Devroye, “The equivalence of weak, strong and complete convergence in L 1 for kernel density estimates,” Annals of Statistics, vol. 11, pp. 896–904, 1983.
L. Devroye, “A note on the L 1 consistency of variable kernel estimates,” Annals of Statistics, vol. 13, pp. 1041–1049, 1985.
L. Devroye, “The kernel estimate is relatively stable,” Probability Theory and Related Fields, vol. 77, pp. 521–536, 1988a.
L. Devroye, “Asymptotic performance bounds for the kernel estimate,” Annals of Statistics, vol. 16, pp. 1162–1179, 1988b.
L. Devroye, “The double kernel method in density estimation,” Annales de l’Institut Henri Poincaré, vol. 25, pp. 533–580, 1989a.
L. Devroye, “Nonparametric density estimates with improved performance on given sets of densities,” Statistics (Mathematische Operationsforschung und Statistik), vol. 20, pp. 357–376, 1989b.
L. Devroye, “Exponential inequalities in nonparametric estimation,” in: Nonparametric Functional Estimation and Related Topics (edited by G. Roussas), pp. 31–44, NATO ASI Series, Kluwer Academic, Dordrecht, 1991.
L. Devroye, “Universal smoothing factor selection in density estimation: Theory and practice (with discussion),” Test, vol. 6, pp. 223–320, 1997.
L. Devroye, A Course in Density Estimation, Birkhäuser-Verlag, Boston, 1987.
L. Devroye and L. Györfi, Nonparametric Density Estimation: The L 1 View, John Wiley, New York, 1985.
L. Devroye and C. S. Penrod, “Distribution-free lower bounds in density estimation,” Annals of Statistics, vol. 12, pp. 1250–1262, 1984.
L. Devroye and M. P. Wand, “On the effect of density shape on the performance of its kernel estimate,” Statistics, vol. 24, pp. 215–233, 1993.
V. A. Epanechnikov, “Nonparametric estimation of a multivariate probability density,” Theory of Probability and its Applications, vol. 14, pp. 153–158, 1969.
P. Hall and M. P. Wand, “Minimizing L 1 distance in nonparametric density estimation,” Journal of Multivariate Analysis, vol. 26, pp. 59–88, 1988.
L. Holmström and J. Klemelä, “Asymptotic bounds for the expected L 1 error of a multivariate kernel density estimator,” Journal of Multivariate Analysis, vol. 40, pp. 245–255, 1992.
E. A. Nadaraya, “On the integral mean square error of some nonparametric estimates for the density function,” Theory of Probability and its Applications, vol. 19, pp. 133–141, 1974.
E. Parzen, “On the estimation of a probability density function and the mode,” Annals of Mathematical Statistics, vol. 33, pp. 1065–1076, 1962.
M. Rosenblatt, “Remarks on some nonparametric estimates of a density function,” Annals of Mathematical Statistics, vol. 27, pp. 832–837, 1956.
M. Rosenblatt, “Curve estimates,” Annals of Mathematical Statistics, vol. 42, pp. 1815–1842, 1971.
C. J. Stone, “An asymptotically optimal window selection rule for kernel density estimates,” Annals of Statistics, vol. 12, pp. 1285–1297, 1984.
G. S. Watson and M. R. Leadbetter, “On the estimation of the probability density, I,” Annals of Mathematical Statistics, vol. 34, pp. 480–491, 1963.
M. Wegkamp, “Quasi universal bandwidth selection for kernel density estimators,” Canadian Journal of Statistics, 2000. To appear.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media New York
About this chapter
Cite this chapter
Devroye, L., Lugosi, G. (2001). The Kernel Density Estimate. In: Combinatorial Methods in Density Estimation. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0125-7_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0125-7_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6527-6
Online ISBN: 978-1-4613-0125-7
eBook Packages: Springer Book Archive